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  • 1
  
  Chapter 01: Mathematical Logic
  
  Subtopics
  
  1.1 Statement
  
  1.2 Logical Connectives, Compound Statements and Truth Tables
  
  1.3 Statement Pattern and Logical Equivalence Tautology, Contradiction
  
  and Contingency
  
  1.4 Quantifiers and Quantified Statements
  
  1.5 Duality
  
  1.6 Negation of Compound Statement
  
  1.7 Algebra of Statements (Some Standard equivalent Statements)
  
  1.8 Application of Logic to Switching Circuits
  
  Mathematical Logic 01
  
  • 2
  
  Std. XII : Perfect Maths - I
  
  Type of Problems Exercise Q. Nos.
  
  Identify the statements and write
  
  down their Truth Value
  
  1.1 Q.1
  
  Miscellaneous Q.1
  
  Express the statements in Symbolic
  
  Form/Write the statement in
  
  Symbolic Form
  
  1.2 Q.1
  
  1.4 Q.1, 2
  
  Miscellaneous Q.5
  
  Write the Truth values of
  
  Statements
  
  1.2 Q.2
  
  1.4 Q.3, 5
  
  1.6 Q.1
  
  Miscellaneous Q.2, 3, 9
  
  Write the Negation of
  
  Statements/Using the Rules of
  
  Negation write the Negation of
  
  Statements
  
  1.3 Q.1
  
  1.8 Q.1, 2, 4
  
  Miscellaneous Q.4, 11, 22
  
  Write the Verbal statement for the
  
  given Symbolic Statement
  
  1.4 Q.6
  
  Miscellaneous Q.6
  
  Converse, Inverse and
  
  Contrapositive of the statement
  
  1.4 Q.4
  
  Miscellaneous Q.19, 21
  
  Using Quantifiers Convert Open
  
  sentences into True statement 1.6 Q.2
  
  Prepare the Truth Table/Find Truth
  
  Values of p and q for given cases
  
  1.4 Q.7
  
  1.5 Q.1
  
  Miscellaneous Q. 12, 15
  
  Examine the statement Patterns
  
  Contingency)
  
  1.5 Q.3
  
  Miscellaneous Q.13, 14, 16
  
  Using Truth Table, Verify Logical
  
  Equivalence
  
  1.5 Q.2
  
  Miscellaneous Q.7, 18
  
  Write Dual of the statement 1.7 Q.1, 2, 3, 4
  
  Algebra of statements (without
  
  using Truth Table verify the Logical
  
  Equivalence)/Rewrite the statement
  
  without using the conditional form
  
  1.8 Q.3
  
  Miscellaneous Q.8, 17, 20
  
  Change the statements in the form if
  
  then Miscellaneous Q.10
  
  Applications of logic to switching
  
  circuits
  
  1.9 Q.1 to 5
  
  Miscellaneous Q.23 to 29
  
  • 3
  
  Chapter 01: Mathematical Logic
  
  Introduction
  
  Mathematics is an exact science. Every mathematical statement must be precise. Hence,
  
  there has to be proper reasoning in every
  
  mathematical proof.
  
  Proper reasoning involves logic. The study of logic helps in increasing ones ability of systematic and
  
  logical reasoning. It also helps to develop the skills
  
  of understanding various statements and their
  
  validity.
  
  Logic has a wide scale application in circuit designing, computer programming etc. Hence, the
  
  study of logic becomes essential.
  
  Statement and its truth value
  
  There are various means of communication viz., verbal, written etc. Most of the communication
  
  involves the use of language whereby, the ideas are conveyed through sentences.
  
  There are various types of sentences such as: i. Declarative (Assertive)
  
  ii. Imperative (A command or a request)
  
  iii. Exclamatory (Emotions, excitement)
  
  iv. Interrogative (Question)
  
  Statement
  
  A statement is a declarative sentence which is either
  
  true or false but not both simultaneously. Statements are denoted by the letters p, q, r.
  
  For example: i. 3 is an odd number.
  
  ii. 5 is a perfect square.
  
  iii. Sun rises in the east.
  
  iv. x + 3 = 6, when x = 3.
  
  Truth Value
  
  A statement is either True or False. The Truth value of a true statement is defined to be T (TRUE) and that of a false statement is defined to be F
  
  (FALSE).
  
  Note: 0 and 1 can also be used for T and F respectively.
  
  Consider the following statements:
  
  i. There is no prime number between 23 and 29.
  
  ii. The Sun rises in the west.
  
  iii. The square of a real number is negative.
  
  iv. The sum of the angles of a plane triangle is
  
  180.
  
  Here, the truth value of statement i. and iv. is T and that of ii. and iii. is F.
  
  Note:
  
  The sentences like exclamatory, interrogative,
  
  imperative etc., are not considered as statements as the truth value for these statements cannot be
  
  determined.
  
  Open sentence
  
  An open sentence is a sentence whose truth can vary
  
  according to some conditions, which are not stated
  
  in the sentence.
  
  Note:
  
  Open sentence is not considered as statement in logic.
  
  For example:
  
  i. x 5 = 20
  
  This is an open sentence as its truth depends
  
  on value of x (if x = 4, it is true and if x 4, it is false).
  
  ii. Chinese food is very tasty.
  
  This is an open sentence as its truth varies
  
  from individual to individual.
  
  Exercise 1.1
  
  State which of the following sentences are
  
  statements, write down the truth value.
  
  i. The Sun is a star.
  
  ii. May God bless you!
  
  iii. The sum of interior angles of a triangle is
  
  180.
  
  iv. Every real number is a complex number.
  
  v. Why are you upset?
  
  vi. Every quadratic equation has two real
  
  roots.
  
  vii. 9 is a rational number.
  
  viii. x2 3x + 2 = 0, implies that x = 1 or x = 2.
  
  ix. The sum of cube roots of unity is one.
  
  x. Please get me a glass of water.
  
  xi. He is a good person.
  
  xii. Two is the only even prime number.
  
  xiii. sin 2 = 2sin cos for all R.
  
  xiv. What a horrible sight it was!
  
  xv. Do not disturb.
  
  xvi. x2 3x 4 = 0, x = 1.
  
  xvii. Can you speak in French?
  
  xviii. The square of every real number is positive.
  
  xix. It is red in colour.
  
  xx. Every parallelogram is a rhombus.
  
  • 4
  
  Std. XII : Perfect Maths - I
  
  Solution:
  
  i. It is a statement which is true, hence its truth value is T.
  
  ii. It is an exclamatory sentence, hence, it is not a statement.
  
  iii. It is a statement which is true, hence its truth
  
  value is T.
  
  iv. It is a statement which is true, hence its truth
  
  value is T.
  
  v. It is an interrogative sentence, hence it is not a statement.
  
  vi. It is a statement which is false, hence its truth value is F.
  
  vii. It is a statement which is false, hence its truth
  
  value is F.
  
  viii. It is a statement which is false, hence its truth
  
  value is F.
  
  ix. It is a statement which is false, hence its truth value is F.
  
  x. It is an imperative sentence, hence it is not a statement.
  
  xi. It is an open sentence, hence it is not a
  
  statement.
  
  xii. It is a statement which is true, hence its truth
  
  value is T.
  
  xiii. It is a statement which is true, hence its truth value is T.
  
  xiv. It is an exclamatory sentence, hence it is not a statement.
  
  xv. It is an imperative sentence, hence it is not a
  
  statement.
  
  xvi. It is a statement which is true, hence its truth
  
  value is T.
  
  xvii. It is an interrogative sentence, hence, it is not a statement.
  
  xviii. It is a statement which is false, hence its truth value is F. (Since, 0 is a real number and
  
  square of 0 is 0 which is neither positive nor
  
  negative).
  
  xix. It is an open sentence, hence it is not a
  
  statement. (The truth of this sentence depends
  
  upon the reference for the pronoun It.)
  
  xx. It is a statement which is false, hence its truth
  
  value is F.
  
  Logical Connectives, Compound Statements and
  
  Truth Tables
  
  Logical Connectives: The words or group of words such as and, or, if .
  
  then, if and only if, not are used to join or connect
  
  two or more simple sentences. These connecting words are called logical connectives.
  
  Compound Statements:
  
  The new statement that is formed by combining two or more simple statements by using logical
  
  connectives are called compound statements.
  
  Component Statements:
  
  The simple statements that are joined using logical
  
  connectives are called component statements.
  
  For example:
  
  Consider the following simple statements,
  
  i. e is a vowel
  
  ii. b is a consonant
  
  These two component statements can be joined by
  
  using the logical connective or as shown below:
  
  e is a vowel or b is a consonant
  
  The above statement is called compound statement formed by using logical connective or.
  
  Truth Table
  
  A table that shows the relationship between truth values of simple statements and the truth values of
  
  compounds statements formed by using these simple
  
  statements is called truth table.
  
  Note:
  
  The truth value of a compoud statement depends
  
  upon the truth values of its component statements.
  
  Logical Connectives
  
  A. AND [ ] (Conjunction):
  
  If p and q are any two statements connected by the word and, then the resulting
  
  compound statement p and q is called
  
  conjunction of p and q which is written in the
  
  symbolic form as p q.
  
  For example:
  
  p: Today is a pleasant day.
  
  q: I want to go for shopping.
  
  The conjunction of above two statements is
  
  p q i.e. Today is a pleasant day and I want to go for shopping.
  
  A conjunction is true if and only if both p and q are true.
  
  Truth table for conjunction of p and q is as
  
  shown below:
  
  p q p q
  
  T T T T F F
  
  F T F F F F
  
  • 5
  
  Chapter 01: Mathematical Logic
  
  Note:
  
  The words such as but, yet, still, inspite, though, moreover are also used to connect the
  
  simple statements.
  
  These words are generally used by replacing and.
  
  B. OR [ ] (Disjunction):
  
  If p and q are any two statements connected by
  
  the word or, then the resulting compound
  
  statement p or q is called disjunction of p and
  
  q which is written in the symbolic form as
  
  p q.
  
  The word or is used in English language in
  
  two distinct senses, exclusive and inclusive.
  
  For example:
  
  i. Rahul will pass or fail in the exam.
  
  ii. Candidate must be graduate or
  
  In eg. (i), or indicates that only one of the
  
  two possibilities exists but not both which is
  
  called exclusive sense of or. In eg. (ii), or
  
  indicates that first or second or both the
  
  possibilities may exist which is called
  
  inclusive sense of or.
  
  A disjunction is false only when both p and q
  
  are false.
  
  Truth table for disjunction of p and q is as
  
  shown below:
  
  p q p q
  
  T T T
  
  T F T
  
  F T T
  
  F F F Exercise 1.2
  
  1. Express the following statements in
  
  symbolic form:
  
  i. Mango is a fruit but potato is a
  
  vegetable.
  
  ii. Either we play football or go for
  
  cycling.
  
  iii. Milk is white or grass is green.
  
  iv. Inspite of physical disability, Rahul
  
  stood first in the class.
  
  v. Jagdish stays at home while Shrijeet and Shalmali go for a movie.
  
  Solution: i. Let p : Mango is a fruit, q : Potato is a vegetable.
  
  The symbolic form of the given statement is
  
  p q.
  
  ii. Let p : We play football, q : We go for cycling.
  
  The symbolic form of the given statement is
  
  p q.
  
  iii. Let p : Milk is white, q : Grass is green.
  
  The symbolic form of the given statement is
  
  p q.
  
  iv. Let p : Rahul has physical disability,
  
  q : Rahul stood first in the class.
  
  The given statement can be considered as Rahul has physical disability and he stood first in the
  
  class.
  
  The symbolic form of the given statement is
  
  p q.
  
  v. Let p : Jagdish stays at home,
  
  q : Shrijeet and Shalmali go for a movie.
  
  The given statement can be considered as Jagdish stays at home and Shrijeet and Shalmali go for a movie.
  
  The symbolic form of the given statement is
  
  p q.
  
  2. Write the truth values of following statements.
  
  i. 3 is a rational number or 3 + i is a
  
  complex number.
  
  ii. Jupiter is a planet and Mars is a star.
  
  iii. 2 + 3 5 or 2 3 < 5
  
  iv. 2 0 = 2 and 2 + 0 = 2
  
  v. 9 is a perfect square but 11 is a prime number.
  
  vi. Moscow is in Russia or London is in France.
  
  Solution:
  
  i. Let p : 3 is a rational number,
  
  q : 3 + i is a complex number.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since, truth value of p is F and that of q is T.
  
  truth value of p q is T
  
  ii. Let p : Jupiter is a planet,
  
  q : Mars is a star.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since, truth value of p is T and that of q is F.
  
  truth value of p q is F
  
  • 6
  
  Std. XII : Perfect Maths - I
  
  iii. Let p : 2 + 3 5,
  
  q : 2 3 < 5.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since, truth value of both p and q is F.
  
  truth value of p q is F
  
  iv. Let p : 2 0 = 2, q : 2 + 0 = 2.
  
  The symbolic form of the given statement is
  
  p q. Since, truth value of p is F and that of q is T.
  
  truth value of p q is F
  
  v. Let p : 9 is a perfect square,
  
  q : 11 is a prime number.
  
  The symbolic form of the given statement is
  
  p q. Since, truth value of both p and q is T.
  
  truth value of p q is T
  
  vi. Let p : Moscow is in Russia, q : London is in France.
  
  The symbolic form of the given statement is
  
  p q. Since, truth value of p is T and that of q is F.
  
  truth value of p q is T
  
  C. Not [~] (Negation): If p is any statement then negation of p
  
  i.e., not p is denoted by ~p. Negation of any simple statement p can also be formed by
  
  writing It is not true that or It is false that,
  
  before p.
  
  For example: p : Mango is a fruit.
  
  ~p : Mango is not a fruit.
  
  Truth table for negation is as shown below:
  
  p ~p
  
  T F
  
  F T
  
  Note: If a statement is true its negation is false and vice-versa.
  
  Exercise 1.3
  
  Write negations of the following statements:
  
  i. Rome is in Italy. ii. 5 + 5 = 10
  
  iii. 3 is greater than 4.
  
  iv. John is good in river rafting.
  
  v. is an irrational number. vi. The square of a real number is positive.
  
  vii. Zero is not a complex number.
  
  viii. Re (z) | z |.
  
  ix. The sun sets in the East.
  
  x. It is not true that the mangoes are inexpensive.
  
  Solution:
  
  i. Rome is not in Italy.
  
  ii. 5 + 5 10
  
  iii. 3 is not greater than 4.
  
  iv. John is not good in river rafting.
  
  v. is not an irrational number.
  
  vi. The square of a real number is not positive.
  
  vii. Zero is a complex number.
  
  viii. Re (z) > |z|.
  
  ix. The sun does not set in the East.
  
  x. It is true that the mangoes are inexpensive.
  
  D. If.then (Implication, ) (Conditional):
  
  If p and q are any two simple statements, then
  
  the compound statement, if p then q, meaning statement p implies statement q or
  
  statement q is implied by statement p, is
  
  called a conditional statement and is denoted
  
  by p q or p q.
  
  Here p is called the antecedent (hypothesis)
  
  and q is called the consequent (conclusion).
  
  For example:
  
  Let p: I travel by train.
  
  q: My journey will be cheaper.
  
  Here the conditional statement is
  
  p q: If I travel by train then my journey
  
  will be cheaper.
  
  Conditional statement is false if and only if
  
  antecedent is true and consequent is false.
  
  Truth table for conditional is as shown
  
  below:
  
  p q p q
  
  T T T
  
  T F F
  
  F T T
  
  F F T
  
  Note: Equivalent forms of the conditional statement
  
  p q:
  
  a. p is sufficient for q.
  
  b. q is necessary for p.
  
  c. p implies q.
  
  d. p only if q.
  
  e. q follows from p.
  
  • 7
  
  Chapter 01: Mathematical Logic
  
  E. Converse, Inverse and Contrapositive
  
  statements:
  
  If p q is given, then its
  
  converse is q p
  
  inverse is ~p ~q
  
  contrapositive is ~q ~p
  
  For example:
  
  Let p : Smita is intelligent.
  
  q : Smita will join Medical.
  
  i. q p: If Smita joins Medical then she
  
  is intelligent.
  
  ii. ~p ~q: If Smita is not intelligent then
  
  she will not join Medical.
  
  iii. ~q ~p: If Smita does not join Medical
  
  then she is not intelligent.
  
  Consider, the following truth table:
  
  p q pq ~p ~q qp ~q~p ~p ~q
  
  T T T F F T T T
  
  T F F F T T F T
  
  F T T T F F T F
  
  F F T T T T T T
  
  From the above table, we conclude that
  
  i. a conditional statement and its
  
  contrapositive are always equivalent.
  
  ii. converse and inverse of the conditional
  
  statement are always equivalent.
  
  F. If and only if (Double Implication, )
  
  (Biconditional):
  
  If p and q are any two statements, then
  
  p if and only if q or p iff q is called the biconditional statement and is denoted by
  
  p q. Here, both p and q are called implicants.
  
  For example: Let p : price increases
  
  q : demand falls
  
  Here the Biconditional statement is
  
  p q : Price increases if and only if demand
  
  falls.
  
  A biconditional statement is true if and only if
  
  both the implicants have same truth value.
  
  Truth table for biconditional is as shown below:
  
  p q p q
  
  T T T T F F
  
  F T F
  
  F F T
  
  Exercise 1.4
  
  1. Express the following in symbolic form.
  
  i. I like playing but not singing.
  
  ii. Anand neither likes cricket nor tennis.
  
  iii. Rekha and Rama are twins.
  
  iv. It is not true that i is a real number.
  
  v. Either 25 is a perfect square or 41 is divisible by 7.
  
  vi. Rani never works hard yet she gets
  
  good marks.
  
  vii. Eventhough it is not cloudy, it is still
  
  raining.
  
  Solution:
  
  i. Let p: I like playing, q: I like singing,
  
  The symbolic form of the given statement is
  
  p ~q.
  
  ii. Let p: Anand likes cricket, q: Anand likes tennis.
  
  The symbolic form of the given statement is
  
  ~p ~q.
  
  iii. In this statement and is combining two nouns
  
  and not two simple statements.
  
  Hence, it is not used as a connective, so given statement is a simple statement which can be
  
  symbolically expressed as p itself.
  
  iv. Let, p : i is a real number.
  
  The symbolic form of the given statement is ~p.
  
  v. Let p : 25 is a perfect square,
  
  q : 41 is divisible by 7.
  
  The symbolic form of the given statement is
  
  p q.
  
  vi. Let p : Rani works hard, q : Rani gets good
  
  marks.
  
  The symbolic form of the given statement is
  
  ~p q.
  
  vii. Let p : It is cloudy, q : It is still raining.
  
  The symbolic form of the given statement is
  
  ~p q.
  
  2. If p: girls are happy, q: girls are playing, express the following sentences in symbolic
  
  form.
  
  i. Either the girls are happy or they are
  
  not playing.
  
  ii. Girls are unhappy but they are
  
  playing.
  
  iii. It is not true that the girls are not
  
  playing but they are happy.
  
  • 8
  
  Std. XII : Perfect Maths - I
  
  Solution:
  
  i. p ~q ii. ~p q
  
  iii. ~(~q p)
  
  3. Find the truth value of the following statements.
  
  i. 14 is a composite number or 15 is a
  
  prime number.
  
  ii. Neither 21 is a prime number nor it is
  
  divisible by 3.
  
  iii. It is not true that 4+3i is a real
  
  number.
  
  iv. 2 is the only even prime number and 5
  
  divides 26.
  
  v. Either 64 is a perfect square or 46 is a
  
  prime number.
  
  vi. 3 + 5 > 7 if and only if 4 + 6 < 10.
  
  Solution:
  
  i. Let p : 14 is a composite number,
  
  q : 15 is a prime number.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since, truth value of p is T and that of q is F.
  
  truth value of p q is T.
  
  ii. Let p: 21 is a prime number,
  
  q: 21 is divisible by 3.
  
  The symbolic form of the given statement is
  
  ~p ~q.
  
  Since, truth value of p is F and that of q is T
  
  truth value of ~p ~q is F.
  
  iii. Let p: 4 + 3i is a real number.
  
  The symbolic form of the given statement is ~p.
  
  Since, truth value of p is F.
  
  truth value of ~p is T.
  
  iv. Let p: 2 is the only even prime number,
  
  q: 5 divides 26.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since, truth value of p is T and that of q is F
  
  truth value of p q is F.
  
  v. Let p: 64 is a perfect square,
  
  q: 46 is a prime number.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since, truth value of p is T and that of q is F.
  
  truth value of p q is T
  
  vi. Let p: 3 + 5 > 7, q: 4 + 6 < 10
  
  The symbolic form of the given statement is
  
  p q .
  
  Since, truth value of p is T and that of q is F.
  
  truth value of p q is F
  
  4. State the converse, inverse and
  
  contrapositive of the following conditional
  
  statements:
  
  i. If it rains then the match will be
  
  cancelled.
  
  ii. If a function is differentiable then it is continuous.
  
  iii. If surface area decreases then the pressure increases.
  
  iv. If a sequence is bounded then it is
  
  convergent.
  
  Solution:
  
  i. Let p : It rains, q : the match will be cancelled.
  
  The symbolic form of the given statement is
  
  p q.
  
  Converse: q p
  
  i.e., If the match is cancelled then it rains.
  
  Inverse: ~p ~q
  
  i.e., If it does not rain then the match will not be
  
  cancelled.
  
  Contrapositive: ~q ~p
  
  i.e. If the match is not cancelled then it does
  
  not rain.
  
  ii. Let p: A function is differentiable,
  
  q: It is continuous.
  
  The symbolic form of the given statement is
  
  p q.
  
  Converse: q p
  
  i.e. If a function is continuous then it is
  
  differentiable.
  
  Inverse: ~p ~q
  
  i.e. If a function is not differentiable then it
  
  is not continuous.
  
  Contrapositive: ~q ~p
  
  i.e. If a function is not continuous then it is
  
  not differentiable.
  
  iii. Let p: Surface area decreases,
  
  q: The pressure increases.
  
  The symbolic form of the given statement is
  
  p q
  
  Converse: q p
  
  i.e. If the pressure increases then the surface
  
  area decreases.
  
  • 9
  
  Chapter 01: Mathematical Logic
  
  Inverse: ~p ~q
  
  i.e. If the surface area does not decrease
  
  then the pressure does not increase.
  
  Contrapositive: ~q ~p
  
  i.e. If the pressure does not increase then the
  
  surface area does not decrease.
  
  iv. Let p: A sequence is bounded,
  
  q: It is convergent.
  
  The symbolic form of the given statement is
  
  p q
  
  Converse: q p
  
  i.e. If a sequence is convergent then it is
  
  bounded.
  
  Inverse: ~p ~q
  
  i.e. If a sequence is not bounded then it is
  
  not convergent.
  
  Contrapositive: ~q ~p
  
  i.e. If a sequence is not convergent then it is
  
  not bounded.
  
  5. If p and q are true and r and s are false
  
  statements, find the truth value of the
  
  following statements:
  
  i. (p q) r
  
  ii. p (r s)
  
  iii. (p s) (q r)
  
  iv. ~ (p r) (q s)
  
  Solution:
  
  Given that p and q are T and r and s are F.
  
  i. (p q) r
  
  (T T) F
  
  T F
  
  T
  
  truth value of the given statement is true.
  
  ii. p (r s)
  
  T (F F)
  
  T T
  
  T
  
  truth value of the given statement is true.
  
  iii. (p s) (q r)
  
  (T F) (T F)
  
  T F
  
  F
  
  truth value of the given statement is false.
  
  iv. ~(p r) (q s)
  
  (T T) (F F)
  
  (T) F
  
  F F
  
  F
  
  truth value of the given statement is false.
  
  6. If p: It is daytime, q: It is warm
  
  Give the compound statements in verbal
  
  form denoted by
  
  i. p q [Oct 14]
  
  ii. p q
  
  iii. p q
  
  iv. q p [Oct 14]
  
  Solution:
  
  i. It is daytime but it is not warm. [1 Mark]
  
  ii. It is daytime or it is warm.
  
  iii. If it is daytime then it is warm.
  
  iv. It is warm if and only if it is daytime.
  
  [1 Mark]
  
  7. Prepare the truth tables for the following:
  
  i. p q ii. p (p q)
  
  iii. p q
  
  Solution:
  
  i. p q
  
  p q ~p ~p q
  
  T T F F
  
  T F F F
  
  F T T T
  
  F F T F
  
  ii. p (p q)
  
  p q p q p (p q)
  
  T T T T
  
  T F T T
  
  F T T T
  
  F F F T
  
  iii. p q
  
  p q ~p ~p q
  
  T T F F
  
  T F F T
  
  F T T T
  
  F F T F
  
  • 10
  
  Std. XII : Perfect Maths - I
  
  Statement Pattern and Logical Equivalence
  
  A. Statement Pattern
  
  Let, p, q, r, be simple statements. A compound statement obtained from these simple statements and by using one or more
  
  connectives , , ~, , is called a statement pattern.
  
  Following points must be noted while preparing truth tables of the statement
  
  patterns:
  
  i. Parentheses must be introduced
  
  wherever necessary.
  
  For example:
  
  ~ (p q) and ~ p q are not the same.
  
  ii. If a statement pattern consists of n statements and m connectives, then truth table consists of 2n rows and
  
  (m + n) columns.
  
  B. Logical equivalence
  
  Two logical statements are said to be equivalent if and only if the truth values in their respective columns in the joint truth table
  
  are identical.
  
  If S1 and S2 are logically equivalent statement patterns, we write
  
  S1 S2.
  
  For example:
  
  To prove: p q ~(p ~q)
  
  [Mar 08]
  
  p q p q ~q (p ~q) ~(p ~q)
  
  T T T F F T
  
  T F F T T F
  
  F T F F T F
  
  F F F T T F
  
  In the above truth table, all the entries in the
  
  columns of p q and ~(p ~q) are identical.
  
  p q ~(p ~q).
  
  Note:
  
  i. ~(p q) ~p ~q (De-Morgans 1st Law)
  
  [Mar 96]
  
  p q ~p ~q p q ~(p q) ~p ~q
  
  T T F F T F F
  
  T F F T T F F F T T F T F F
  
  F F T T F T T
  
  ii. ~(p q) ~p ~q (De-Morgans 2nd Law)
  
  p q ~p ~q p q ~(p q) ~p ~q
  
  T T F F T F F T F F T F T T
  
  F T T F F T T
  
  F F T T F T T
  
  iii. p q (~p) q
  
  p q ~p p q ~p q
  
  T T F T T
  
  T F F F F F T T T T
  
  F F T T T
  
  iv. p q (p q) (q p)
  
  [Mar 98, Oct 00, 01,04]
  
  p q p q q p p q (p q) (q p)
  
  T T T T T T
  
  T F F T F F
  
  F T T F F F F F T T T T
  
  Tautology
  
  A statement pattern having truth value always T, irrespective of the truth values of its component
  
  statement is called Tautology.
  
  For example, consider (p q) (q p)
  
  p q p q q p (p q) (q p)
  
  T T T T T
  
  T F F F T
  
  F T F F T
  
  F F T T T
  
  In the above truth table, all the entries in the last column are T.
  
  The given statement pattern is a tautology.
  
  A statement pattern having truth value always F, irrespective of the truth values of its component
  
  identical
  
  identical
  
  identical
  
  identical
  
  l
  
  • 11
  
  Chapter 01: Mathematical Logic
  
  For example, consider p ~ p
  
  p ~ p p ~ p
  
  T F F
  
  F T F
  
  In the above truth table, all the entries in the last column
  
  are F.
  
  The given statement pattern is a contradiction.
  
  Contingency
  
  A statement pattern which is neither a tautology nor
  
  For example, consider (p q) ~(p ~ q)
  
  p q p q ~q p ~q ~(p ~q) (p q)
  
  ~(p ~q)
  
  T T T F F T T
  
  T F F T T F F
  
  F T F F T F F
  
  F F T T T F F
  
  In the above truth table, the entries in the last column are a combination of T and F.
  
  The given statement pattern is neither a
  
  tautology nor a contradiction, it is a
  
  contingency.
  
  Exercise 1.5
  
  1. Prepare the truth table of the following statement patterns:
  
  i. [(p q) q] p
  
  ii. (p q) (~p)
  
  iii. (p q) (~p q)
  
  iv. (p r) (q p)
  
  v. (p ~q) (r p)
  
  Solution:
  
  i. [(p q) q] p
  
  p q p q (p q) q [(p q) q] p
  
  T
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  T
  
  T
  
  F
  
  T
  
  F
  
  T
  
  T
  
  F
  
  T
  
  Note:
  
  Here, the statement pattern consists of 2
  
  statements and 3 connectives. Therefore, the truth table has,
  
  Number of rows = 22 = 4
  
  and number of columns = (2+3) = 5.
  
  ii. (p q) (~p)
  
  p q p q ~p (p q) (~p)
  
  T
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  F
  
  F
  
  F
  
  F
  
  T
  
  T
  
  F
  
  T
  
  T
  
  T
  
  iii. (p q) (~p q)
  
  p q p q ~p ~p q (p q)
  
  (~p q)
  
  T
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  T
  
  F
  
  F
  
  T
  
  T
  
  T
  
  F
  
  T
  
  T
  
  T
  
  T
  
  T
  
  T
  
  iv. (p r) (q p)
  
  p q r p r q p (p r)
  
  (q p)
  
  T
  
  T
  
  T
  
  T
  
  F
  
  F
  
  F
  
  F
  
  T
  
  T
  
  F
  
  F
  
  T
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  T
  
  T
  
  F
  
  F
  
  F
  
  F
  
  T
  
  T
  
  T
  
  F
  
  F
  
  F
  
  F
  
  F
  
  F
  
  T
  
  v. (p ~q) (r p)
  
  p q r ~q p ~ q r p (p ~q)
  
  (r p)
  
  T
  
  T
  
  T
  
  T
  
  F
  
  F
  
  F
  
  F
  
  T
  
  T
  
  F
  
  F
  
  T
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  F
  
  F
  
  T
  
  T
  
  F
  
  F
  
  T
  
  T
  
  T
  
  T
  
  T
  
  T
  
  F
  
  F
  
  T
  
  T
  
  T
  
  F
  
  T
  
  F
  
  F
  
  F
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  T
  
  F
  
  F
  
  2. Using truth tables, prove the following logical equivalences:
  
  i. (p q) ~ (p ~ q)
  
  ii. p q (p q) (~p ~q)
  
  [Mar 13; Oct 15]
  
  iii. (p q) r p (q r) [Oct 14]
  
  iv. p (q r) (p q) (p r)
  
  • 12
  
  Std. XII : Perfect Maths - I
  
  Solution:
  
  i.
  
  1 2 3 4 5 6
  
  p q p q ~q p ~q ~ (p ~q)
  
  T
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  F
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  T
  
  T
  
  T
  
  F
  
  F
  
  F
  
  The entries in the columns 3 and 6 are identical.
  
  (p q) ~ (p ~q)
  
  ii.
  
  1 2 3 4 5 6 7 8
  
  p q p q ~p ~q p q ~p ~q (p q)
  
  (~p ~q)
  
  T T T F F T F T
  
  T F F F T F F F
  
  F T F T F F F F
  
  F F T T T F T T
  
  The entries in columns 3 and 8 are identical.
  
  p q (p q) (~p ~q) [1 Mark]
  
  [1 mark each for column 3 and column 8]
  
  iii.
  
  1 2 3 4 5 6 7
  
  p q r p q q r (p q) r p
  
  (q r)
  
  T T T T T T T
  
  T T F T F F F
  
  T F T F T T T
  
  T F F F T T T
  
  F T T F T T T
  
  F T F F F T T
  
  F F T F T T T
  
  F F F F T T T
  
  The entries in the columns 6 and 7 are identical.
  
  (p q) r p (q r) [1 Mark]
  
  [1 mark each for column 6 and column 7]
  
  iv.
  
  1 2 3 4 5 6 7 8
  
  p q r q r p (q r) p q p r (pq) (pr)
  
  T T T T T T T T
  
  T T F F T T T T
  
  T F T F T T T T
  
  T F F F T T T T
  
  F T T T T T T T
  
  F T F F F T F F
  
  F F T F F F T F
  
  F F F F F F F F
  
  The entries in the columns 5 and 8 are identical
  
  p (q r) (p q) (p r)
  
  3. Using truth tables examine whether the
  
  following statement patterns are tautology,
  
  i. (p ~ q) (p q) [Mar 13]
  
  ii. (~p q) (q p)
  
  iii. (p q) (p r)
  
  iv. [(p q) r] [p (q r)]
  
  v. (p q) (p r)
  
  Solution:
  
  i.
  
  1 2 3 4 5 6
  
  p q ~q p ~q p q (p ~ q)
  
  (p q)
  
  T T F F T F
  
  T F T T F F
  
  F T F F T F F F T F T F
  
  In the above truth table, all the entries in the last
  
  column are F.
  
  (p ~q) (p q) is a contradiction. [1 Mark]
  
  [1 mark each for column 5 and column 6]
  
  ii.
  
  p q ~p ~p q q p (~p q)
  
  (q p)
  
  T T F F T F
  
  T F F F T F
  
  F T T T F F
  
  F F T F T F
  
  In the above truth table, all the entries in the last
  
  column are F.
  
  (~p q) (q p) is a contradiction.
  
  • 13
  
  Chapter 01: Mathematical Logic
  
  iii.
  
  p q r p q p r (p q) (p r)
  
  T T T T T T
  
  T T F T F T
  
  T F T F T T
  
  T F F F F F
  
  F T T F F F
  
  F T F F F F
  
  F F T F F F
  
  F F F F F F
  
  In the above truth table, the entries in the last column are a combination of T and F.
  
  (p q) (p r) is a contingency.
  
  iv.
  
  p q r p q q r (p q)
  
  r
  
  p (q r) [(pq)r]
  
  [p(q r)]
  
  T T T T T T T T
  
  T T F T T T T T
  
  T F T T T T T T
  
  T F F T F T T T
  
  F T T T T T T T
  
  F T F T T T T T
  
  F F T F T T T T
  
  F F F F F F F T
  
  In the above truth table, all the entries in the last column are T.
  
  [(p q) r] [p (q r)] is a tautology.
  
  v.
  
  p q r p q p r (p q) (p r)
  
  T T T T T T
  
  T T F T T T
  
  T F T T T T
  
  T F F T T T
  
  F T T T T T
  
  F T F T F F
  
  F F T F T F
  
  F F F F F F
  
  In the above truth table, the entries in the last
  
  column are a combination of T and F.
  
  (p q) (p r) is a contingency.
  
  Quantifiers and Quantified Statements
  
  Quantifiers
  
  Quantifiers are the symbols used to denote a group of words or a phrase. Generally, two types of
  
  quantifiers are used. They are as follows:
  
  i. Universal Quantifier:
  
  The symbol stands for all values of and is known as universal quantifier.
  
  For example, Consider A = {1, 2, 3}
  
  Let p: x A, x < 4
  
  Here, the statement p uses the quantifier for
  
  all().
  
  This statement is true if and only if each and
  
  every element of set A satisfies the condition
  
  x < 4 and is false otherwise.
  
  Here, the given statement is true for all the
  
  elements of set A, as 1, 2, 3 satisfy the
  
  condition, x A, x < 4.
  
  ii. Existential Quantifier:
  
  The symbol stands for there exists and is
  
  known as existential quantifier.
  
  For example, Consider A = {4, 14, 66, 70}.
  
  Let p: x A such that x is an odd number.
  
  Here, the statement p uses the quantifier there
  
  exists ().
  
  This statement is true if atleast one element of
  
  set A satisfies the condition x is an odd
  
  number and is false otherwise.
  
  Here, the given statement is false as none of
  
  the elements of set A satisfy the condition,
  
  x A such that x is an odd number.
  
  Quantified statement
  
  The statement containing quantifiers is known as
  
  quantified statement. Generally, an open sentence
  
  with a quantifier becomes a statement and is called quantified statement.
  
  For example:
  
  Use quantifiers to convert open sentence x + 2 < 4
  
  into a statement.
  
  Solution:
  
  x N such that x + 2 < 4, is a true
  
  statement, since x = 1 N satisfies x + 2 < 4.
  
  Exercise 1.6
  
  1. If A = {3, 4, 6, 8}, determine the truth value
  
  of each of the following:
  
  i. x A, such that x + 4 = 7
  
  ii. x A, x + 4 < 10.
  
  iii. x A, x + 5 13.
  
  iv. x A, such that x is odd.
  
  v. x A, such that (x 3) N
  
  • 14
  
  Std. XII : Perfect Maths - I
  
  Solution:
  
  i. Since x = 3 A, satisfies x + 4 = 7
  
  the given statement is true.
  
  Its truth value is T.
  
  ii. Since x = 6, 8 A, do not satisfy x + 4 < 10,
  
  the given statement is false.
  
  Its truth value is F
  
  iii. Since x = 3, 4, 6 A, do not satisfy x + 5 13,
  
  the given statement is false.
  
  Its truth value is F.
  
  iv. Since x = 3 A, satisfies the given statement,
  
  the given statement is true.
  
  Its truth value is T.
  
  v. Since x = 4, 6, 8 A, satisfy (x 3) N,
  
  the given statement is true.
  
  Its truth value is T. 2. Use quantifiers to convert each of the
  
  following open sentences defined on N, into
  
  a true statement: i. x2 = 25 ii. 2x + 3 < 15
  
  iii. x 3 = 11 iv. x2 + 1 5
  
  v. x2 3x + 2 = 0 Solution:
  
  i. x N, such that x2 = 25.
  
  This is a true statement since x = 5 N satisfies x2 = 25.
  
  ii. x N, such that 2x + 3 < 15. This is a true statement since x = 1, 2, 3, 4, 5
  
  N satisfy 2x + 3 < 15.
  
  iii. x N such that x 3 = 11.
  
  This is a true statement since x = 14 N
  
  satisfies x 3 = 11.
  
  iv. x N, such that x2 + 1 5.
  
  This is a true statement since x = 1, 2 N
  
  satisfy x2 + 1 5.
  
  v. x N such that x2 3x + 2 = 0.
  
  This is a true statement since x = 1, 2 N
  
  satisfy x2 3x + 2 = 0.
  
  Duality
  
  Two compound statements S1 and S2 are said to be duals of each other, if one can be obtained from the
  
  other by interchanging and and vice-versa.
  
  The connectives and are duals of each other. Also, a dual is obtained by replacing t by c and c by
  
  t, where t denotes tautology and c denotes
  
  Remarks:
  
  i. The symbol is not changed while finding
  
  the dual.
  
  ii. Dual of a dual is the statement itself.
  
  iii. The special statements t (tautology) and c
  
  (contradiction) are duals of each other.
  
  iv. T is changed to F and vice-versa.
  
  Principle of Duality:
  
  If a compound statement S1 contains only ~, and
  
  and statement S2 arises from S1 by replacing by
  
  and by , then S1 is a tautology if and only if S2 is a contradiction.
  
  Exercise 1.7
  
  1. Write the duals of the following statements:
  
  i. (p q) r
  
  ii. T (p q)
  
  iii. p [q (p q) r]
  
  Solution:
  
  i. (p q) r
  
  ii. F (p q)
  
  iii. p [~q (p q) ~r]
  
  2. Write the dual statement of each of the
  
  following compound statements:
  
  i. Vijay and Vinay cannot speak Hindi.
  
  ii. Ravi or Avinash went to Chennai.
  
  iii. Madhuri has curly hair and brown eyes. [Mar 14]
  
  Solution:
  
  i. Vijay or Vinay cannot speak Hindi.
  
  ii. Ravi and Avinash went to Chennai.
  
  iii. Madhuri has curly hair or brown eyes. [1 Mark]
  
  3. Write the duals of the following statements:
  
  i. (p q) r p (q r)
  
  ii. p (q r) (p q) (p r)
  
  Solution:
  
  i. (p q) r p (q r)
  
  ii. p (q r) (p q) (p r)
  
  4. Write duals of each of the following statements where t is a tautology and c is a contradiction.
  
  i. p q c ii. ~ p (q c)
  
  iii. (p t) (c ~ q)
  
  Solution:
  
  i. p q t ii. ~ p (q t)
  
  iii. (p c) (t ~ q)
  
  • 15
  
  Chapter 01: Mathematical Logic
  
  Negation of compound statement
  
  i. Negation of conjunction:
  
  Negation of the conjunction of two simple
  
  statements is the disjunction of their negations.
  
  i.e. ~(p q) ~p ~q
  
  ii. Negation of disjunction:
  
  Negation of the disjunction of two simple statements is the conjunction of their
  
  negations.
  
  i.e. ~(p q) ~p ~q
  
  iii. Negation of negation:
  
  The negation of negation of a simple
  
  statement is the statement itself.
  
  i.e. If p is a simple statement then (p) p
  
  iv. Negation of conditional (implication)
  
  statement:
  
  The negation of a conditional statement p q is p but not q.
  
  i.e. ~(p q) p ~q
  
  v. Negation of biconditional (double
  
  implication) statement:
  
  The negation of a biconditional statement
  
  p q is the negation of p q or q p.
  
  i.e. ~(p q) (p ~q) (q ~p)
  
  Note:
  
  Negation of statement pattern involving one or
  
  more simple statements p, q, r, . and one or
  
  more connectives , or is obtained by
  
  replacing by , by . Statements p, q, r
  
  are replaced by their negations p, q, r etc. and vice-versa.
  
  vi. Negation of a quantified statement:
  
  While finding the negations of quantified
  
  statements, the word all is replaced by
  
  some and for every is replaced by there
  
  exists and vice-versa.
  
  Algebra of Statements
  
  Some standard equivalent statements:
  
  a. Idempotent Law:
  
  i. p p p
  
  ii. p p p
  
  b. Commutative Law:
  
  i. p q q p
  
  ii. p q q p
  
  c. Associative Law:
  
  i. (p q) r p (q r) p q r
  
  ii. (p q) r p (q r) p q r
  
  d. Distributive Law:
  
  i. p (q r) (p q) (p r)
  
  ii. p (q r) (p q) (p r)
  
  e. Identity Law:
  
  i. p F p
  
  ii. p F F
  
  iii. p T T
  
  iv. p T p
  
  f. Complement Law:
  
  i. p ~p T
  
  ii. p ~p F
  
  g. Involution Law:
  
  i. ~T F
  
  ii. ~F T
  
  iii. ~(~p) p
  
  h. DeMorgans Law:
  
  i. ~(p q) ~p ~q
  
  ii. ~(p q) ~p ~q
  
  i. Absorption Law:
  
  i. p (p q) p
  
  ii. p (p q) p
  
  j. Conditional Law:
  
  i. p q ~p q
  
  ii. p q (~p q) (~q p)
  
  Exercise 1.8
  
  1. Write the negations of following statements.
  
  i. All equilateral triangles are isosceles.
  
  ii. Some real numbers are not complex numbers.
  
  iii. Every student has paid the fees.
  
  iv. n N, n + 1 > 2.
  
  v. x N, x2 + x is even number.
  
  vi. n N, such that n2 = n.
  
  vii. x R, such that x2 < x.
  
  viii. All students of this college live in the
  
  hostel.
  
  ix. Some continuous functions are
  
  differentiable.
  
  x. Democracy survives if the leaders are
  
  not corrupt.
  
  • 16
  
  Std. XII : Perfect Maths - I
  
  xi. The necessary and sufficient condition
  
  for a person to be successful is to be honest.
  
  unequal roots.
  
  Solution:
  
  i. Some equilateral triangles are not isosceles.
  
  ii. All real numbers are complex numbers.
  
  iii. Some students have not paid the fees.
  
  iv. n N, such that n + 1 2.
  
  v. x N, such that x2 + x is not an even number.
  
  vi. n N, n2 n.
  
  vii. x R, x2 x.
  
  viii. Some students of this college do not live in the
  
  hostel.
  
  ix. All continuous functions are not differentiable.
  
  x. Let, p: The leaders are not corrupt.
  
  q: Democracy survives.
  
  The given statement is of the form, p q
  
  Its negation is of the form p q.
  
  i.e. the leaders are not corrupt, but democracy does not survive.
  
  xi. Let, p: A person is successful.
  
  q: A person is honest.
  
  The given statement is of the form, p q
  
  Its negation is of the form,
  
  (p q) (q p)
  
  i.e. a person is successful, but he is not honest
  
  or a person is honest, but he is not successful.
  
  xii. All quadratic equations have equal roots.
  
  2. Using the rules of negation, write the
  
  negations of the following.
  
  i. p (q r) ii. (~p q) r
  
  iii. (p q) (p q)
  
  Solution:
  
  i. ~[p (q r)]
  
  ~p ~(q r) .(Negation of conjunction)
  
  ~p (q ~r) .(Negation of implication)
  
  ii. ~[(~p q) r]
  
  ~(~p q) ~r ....(Negation of conjunction)
  
  [~(~p) ~q] ~r
  
  .... (Negation of disjunction)
  
  (p ~q) ~r .(Negation of negation)
  
  iii. ~[(~p ~q) (p ~q)]
  
  ~(~p ~q) ~(p ~q)
  
  (Negation of disjunction)
  
  [~(~p) ~ (~q)] [~p ~(~q)]
  
  ....(Negation of conjunction)
  
  (p q) (~p q) .(Negation of negation)
  
  3. Without using truth tables, show that
  
  i. p (q p) p q
  
  ii. (p q) (p q) (q r) q r
  
  Solution:
  
  i. L.H.S. = p (q ~p)
  
  (p q) (p ~p)
  
  (Distributive law)
  
  (p q) F .(Complement law)
  
  p q .(Identity law)
  
  L.H.S. = R.H.S.
  
  p (q p) p q
  
  ii. L.H.S . = (p q) (~p q) (~q r)
  
  [(p ~p) q] (~q r)
  
  (Associative and distributive law)
  
  (T q) (~q r)
  
  ....(Complement law)
  
  q (~q r)
  
  ....(Identity law)
  
  (q ~q) (q r)
  
  ....(Distributive law)
  
  T (q r) ....(Complement law)
  
  q r ....(Identity law)
  
  L.H.S. = R.H.S.
  
  (p q) (p q) (q r) q r
  
  4. Form the negations of the following statements
  
  by giving justification.
  
  i. (p q) (~p r)
  
  ii. (q ~r) (p q)
  
  Solution:
  
  i. ~[(p q) (~p r)]
  
  (p q) ~(~p r)
  
  . (Negation of implication)
  
  (p q) [~(~p) ~r]
  
  . (Negation of disjunction)
  
  (p q) (p ~r)
  
  . (Negation of negation) ii. ~[(q ~r) (p q)]
  
  ~(q ~r) ~(p q)
  
  .(Negation of conjunction)
  
  [~q ~(~r)] (~p ~q)
  
  . (Negation of disjunction)
  
  • 17
  
  Chapter 01: Mathematical Logic
  
  (~q r) (~p ~q)
  
  . (Negation of negation)
  
  (~q r) (~q ~p) .(Commutative law)
  
  ~q (r ~p) .(Distributive law)
  
  Application of Logic to Switching Circuits
  
  The working of an electric switch is similar to a
  
  logical statement which has exactly two outcomes,
  
  namely, T or F. A switch also has two outcomes or
  
  results (current flows and current does not flow)
  
  depending upon the status of the switch i.e. (ON or
  
  OFF). This analogy is very useful in solving
  
  problems of circuit design with the help of logic.
  
  Switch
  
  An electric switch is a two state device used for
  
  turning a current on or off.
  
  As shown in the above figures, if the switch is on
  
  i.e., circuit is closed, current passes through the
  
  circuit and vice-versa.
  
  Consider a simple circuit having a switch S, a
  
  battery and a lamp L. When the switch S is closed (i.e., ON, current is flowing through the
  
  circuit), the lamp glows (is on). Similarly, when the
  
  swtich is open (i.e., OFF, current is not flowing through the circuit), the lamp does not glow (is off).
  
  Thus, if p is a statement the switch is closed and if
  
  l is a statement the lamp glows then p is equivalent
  
  to l i.e p l.
  
  Note:
  
  i. ~p means the switch is open. In this case,
  
  the lamp will not glow and thus p l.
  
  ii. If a switch is ON then its truth value is T or
  
  1 and if the switch is OFF, its truth value is
  
  F or 0.
  
  If there are two switches, then they can be
  
  connected in the following ways:
  
  i. Switches S1 and S2 connected in series:
  
  Let p: the switch S1 is closed
  
  q: the switch S2 is closed
  
  l: the lamp glows
  
  In this case, the lamp glows, if and only if
  
  both the switches are closed.
  
  Thus we have, p q l.
  
  Input Output (switching) table:
  
  p q p q
  
  T (1) T (1) T (1)
  
  T (1) F (0) F (0)
  
  F (0) T (1) F (0)
  
  F (0) F (0) F (0)
  
  ii. Switches S1 and S2 are in parallel
  
  Let p : the switch S1 is closed
  
  q : the switch S2 is closed
  
  l : the lamp glows
  
  In this case, the lamp glows, if at least one of the switches is closed.
  
  Thus, we have p q l,
  
  Input output (switching) table:
  
  p q p q
  
  T (1) T (1) T (1)
  
  T (1) F (0) T (1)
  
  F (0) T (1) T (1)
  
  F (0) F (0) F (0)
  
  The above two networks can be combined
  
  to form a complicated network as shown below:
  
  S
  
  closed (On) switch
  
  S
  
  open (Off ) switch
  
  S1
  
  L
  
  S2
  
  S3
  
  Lamp Battery
  
  S
  
  L
  
  S1
  
  L
  
  S2
  
  S1
  
  L S2
  
  • 18
  
  Std. XII : Perfect Maths - I
  
  Let p : The switch S1 is closed
  
  q : The switch S2 is closed
  
  r : The switch S3 is closed
  
  l : The lamp glows
  
  In this case, the lamp glows, if S1 and S2 both are
  
  closed or if S3 is closed.
  
  Thus we have, (p q) r l.
  
  Note: i. If two or more switches in a circuit are
  
  open or closed simultaneously, then they are denoted by same letter and are called
  
  equivalent switches.
  
  ii. Any two switches in a circuit having opposite states are called
  
  complementary switches.
  
  For example, if S1 and S2 are the two
  
  switches such that when S1 is closed, S2 is open and vice-versa, then the switches
  
  S1 and S2 are called complementary
  
  switches and S2 is denoted as 1S . In
  
  such a situation, one of them is
  
  considered as p and the other as ~p or p.
  
  iii. Two circuits are called equivalent if
  
  output of the two circuits is always same.
  
  iv. A circuit is called simpler if it contains
  
  lesser number of switches.
  
  Example :
  
  Express the following circuit in the symbolic form:
  
  Solution:
  
  Let p : The switch S1 is closed.
  
  q : The switch S2 is closed.
  
  r : The switch S3 is closed.
  
  p : The switch 1S is closed.
  
  r: The switch 3S is closed.
  
  l : Lamp L is on.
  
  The lamp L is on if and only if
  
  i. Switch S1 and Switch S2 are closed.
  
  (Q S1 and S2 are in series.)
  
  or ii. Switch 1S and
  
  3S are closed.
  
  (Q1S and
  
  3S are in series)
  
  and iii. Switch S1 or S2 or S3 are closed.
  
  (Q S1, S2, S3 are in parallel)
  
  Symbolic form of the given circuit is,
  
  [(p q) ( p r)] (p q r) l
  
  Generally l is not written and therefore the
  
  symbolic form is
  
  [(p q) ( p r)] (pq r)
  
  Exercise 1.9
  
  1. Represent the following circuits
  
  symbolically and write the input-output or switching table.
  
  i.
  
  ii.
  
  iii.
  
  Solution:
  
  i. Let p: The switch S1 is closed.
  
  q: The switch S2 is closed.
  
  ~p: The switch 1S is closed or the switch S1 is
  
  open.
  
  ~q: The switch 2S is closed or the switch S2 is
  
  open.
  
  The symbolic form of the given circuit is
  
  (p q) (~p ~q)
  
  S1
  
  S2
  
  S3
  
  L
  
  S1 S2
  
  1S
  
  3S
  
  1S
  
  S1
  
  S2
  
  2S
  
  L
  
  L
  
  1S
  
  2S
  
  S3
  
  S1 S2
  
  S3S1
  
  S2
  
  S2
  
  L
  
  1S
  
  • 19
  
  Chapter 01: Mathematical Logic
  
  Input-output Table:
  
  p q ~p ~q p q ~p ~q (p q)
  
  (~p ~q)
  
  1 1 0 0 1 0 1
  
  1 0 0 1 1 0 1
  
  0 1 1 0 1 0 1
  
  0 0 1 1 0 1 1
  
  Switching Table:
  
  p q ~p ~q p q ~p ~q (p q) (~p ~q)
  
  T T F F T F T
  
  T F F T T F T
  
  F T T F T F T
  
  F F T T F T T
  
  ii. Let p: The switch S1 is closed.
  
  q: The switch S2 is closed.
  
  r: The switch S3 is closed.
  
  ~p: The switch 1S is closed or the switch S1 is
  
  open.
  
  ~q: The switch 2S is closed or the switch S2 is
  
  open.
  
  The symbolic form of the given circuit is
  
  [(p q) (~p ~q)] r.
  
  Let a (p q), b (p q)
  
  Input-output Table: p q r ~p ~q a b a b (a b) r
  
  1 1 1 0 0 1 0 1 1
  
  1 1 0 0 0 1 0 1 0
  
  1 0 1 0 1 0 0 0 0
  
  1 0 0 0 1 0 0 0 0
  
  0 1 1 1 0 0 0 0 0
  
  0 1 0 1 0 0 0 0 0
  
  0 0 1 1 1 0 1 1 1
  
  0 0 0 1 1 0 1 1 0
  
  Switching table:
  
  p q r ~p ~q a b a b (a b) r
  
  T T T F F T F T T
  
  T T F F F T F T F
  
  T F T F T F F F F
  
  T F F F T F F F F
  
  F T T T F F F F F
  
  F T F T F F F F F
  
  F F T T T F T T T
  
  F F F T T F T T F
  
  iii. Let p: The switch S1 is closed
  
  q: The switch S2 is closed
  
  r: The switch S3 is closed
  
  ~p: The switch 1S is closed or the switch S1
  
  is open.
  
  The symbolic form of the given circuit is
  
  (p q) q (r ~p)
  
  Input- output Table:
  
  p q r ~p p q (p q)
  
  q (r ~p)
  
  (p q)
  
  q
  
  (r ~p)
  
  1 1 1 0 1 1 1 1
  
  1 1 0 0 1 1 0 0
  
  1 0 1 0 1 0 1 0
  
  1 0 0 0 1 0 0 0
  
  0 1 1 1 1 1 1 1
  
  0 1 0 1 1 1 1 1
  
  0 0 1 1 0 0 1 0
  
  0 0 0 1 0 0 1 0
  
  Switching table:
  
  p q r ~p p q (p q)
  
  q r ~p
  
  (p q) q
  
  (r ~p)
  
  T T T F T T T T
  
  T T F F T T F F
  
  T F T F T F T F
  
  T F F F T F F F
  
  F T T T T T T T
  
  F T F T T T T T
  
  F F T T F F T F
  
  F F F T F F T F
  
  2. Construct the switching circuits of the following statements.
  
  i. [p (~p q)] [(q r) p]
  
  [Mar 15]
  
  ii. (p q r) [p (q r)]
  
  iii. [(p r) (~q r)] (~p ~r)
  
  Solution:
  
  i. Let p: The switch S1 is closed
  
  q: The switch S2 is closed.
  
  r: The switch S3 is closed.
  
  ~p: The switch 1S is closed or the switch S1 is
  
  open.
  
  ~q: The switch 2S is closed or the switch S2 is
  
  open.
  
  ~r : The switch 3S is closed or the switch S3 is
  
  open. [1 Mark]
  
  • 20
  
  Std. XII : Perfect Maths - I
  
  Consider the given statement,
  
  [p ( p q )] [(q r) p].
  
  p (p q): represents that switch S1 is
  
  connected in parallel with the series combination
  
  of 1S and S2.
  
  (q r ) p : represents that switch 1S is
  
  connected in parallel with the series combination
  
  of 2S and S3.
  
  Therefore, [p (p q )] [(q r) p]
  
  represents that the circuits corresponding to [p
  
  ( p q )] and [(q r) p] are connected in parallel with each other. [1 Mark]
  
  Switching Circuit corresponding to the given
  
  statement is:
  
  ii. Let p : The switch S1 is closed.
  
  q : The switch S2 is closed.
  
  r : The switch S3 is closed.
  
  ~p : The switch 1S is closed or the switch S1
  
  is open.
  
  ~r : The switch 3S is closed or the switch
  
  S3 is open.
  
  Consider the given statement,
  
  (p q r) [ p (q r)]
  
  p q r: represents that switches S1, S2, S3
  
  are connected in series.
  
  p (q r): represents that the series
  
  combination of S2 and 3S is connected in
  
  parallel with 1S .
  
  Therefore, (p q r) [ p (q r) represents that the circuits corresponding to
  
  (p q r) and [ p (q r)] are connected in parallel with each other.
  
  Switching circuit corresponding to the given statement is:
  
  iii. Let p: The switch S1 is closed
  
  q: The switch S2 is closed.
  
  r: The switch S3 is closed.
  
  ~p: The switch 1S is closed or the switch S1 is
  
  open.
  
  ~q: The switch 2S is closed or the switch S2 is
  
  open.
  
  ~r : The switch 3S is closed or the switch S3 is
  
  open. Consider the given statement,
  
  [(p r) (q r)] (p r)
  
  (p r) (q r): represents that the series combination of S1 and S3 and series
  
  combination of 2S and
  
  3S are connected in
  
  parallel with each other.
  
  p r : represents that switches 1S and
  
  3S
  
  are connected in series.
  
  Therefore, [(p r) (q r)] ( p r) represents that the circuits corresponding to
  
  [(p r) (q r)] and (p r) are connected in parallel with each other.
  
  Switching Circuit corresponding to the given statement is:
  
  3. Give an alternative arrangement for the
  
  following circuit, so that the new circuit has two switches only. Also write the switching
  
  table. [July 16]
  
  L
  
  3S
  
  1S
  
  S2
  
  S1 S2 S3
  
  3S
  
  S1 S3
  
  0
  
  2S
  
  1S
  
  L
  
  3S
  
  L
  
  S1
  
  1S
  
  2S
  
  2S
  
  1S S2
  
  L
  
  1S S2
  
  S1
  
  2S
  
  S3
  
  1S
  
  [1 Mark]
  
  • 21
  
  Chapter 01: Mathematical Logic
  
  Solution:
  
  Let p: The switch S1 is closed.
  
  q: The switch S2 is closed.
  
  ~p: The switch 1S is closed or the switch S1 is open.
  
  ~q: The switch 2S is closed or the switch S2 is open.
  
  symbolic form of the given circuit is
  
  (p ~q) (~p q) (~p ~q) [1 Mark]
  
  (pq) [p(qq)] (Associative and Distributive law)
  
  (pq) (pT) .(Complement law)
  
  (pq) p ....(Identity law)
  
  (p ~p) (~q p) ....(Distributive law)
  
  [1 Mark]
  
  (p ~p) (~p q) .(Commutative law)
  
  T(~p ~q) .( Complement law)
  
  pq .(Identity law)
  
  [1 Mark]
  
  The alternative arrangement for given circuit is:
  
  [1 Mark]
  
  Switching Table:
  
  p q ~p ~q p ~q ~p q ~p ~q
  
  (p q)
  
  (p q)
  
  (p q)
  
  T T F F F F F F
  
  T F F T T F F T
  
  F T T F F T F T
  
  F F T T F F T T
  
  4. Find
  
  i. symbolic form
  
  ii. switching table and
  
  iii. draw simplified switching circuit for
  
  the following switching circuit.
  
  Solution:
  
  i. Let p: The switch S1 is closed.
  
  q: The switch S2 is closed.
  
  r: The switch S3 is closed.
  
  ~p: The switch 1S is closed or the switch
  
  S1 is open.
  
  ~q: The switch 2S is closed or the switch
  
  S2 is open.
  
  The symbolic form of the given circuit is
  
  (p q) (~p q) (r ~q)
  
  ii. Switching table:
  
  Let a (p q) (p q) (r q)
  
  p q r pq p q pq rq a
  
  T T T T F F F F T
  
  T T F T F F F F T
  
  T F T F F T F T T
  
  T F F F F T F F F
  
  F T T F T F T F T
  
  F T F F T F T F T
  
  F F T F T T F T T
  
  F F F F T T F F F
  
  iii. For simplified switching circuit,
  
  Consider (p q) (~p q) (r ~q)
  
  [(p ~p) q] (r ~q)
  
  ....(Associative and Distributive law)
  
  (T q) (r ~q)....(Complement law)
  
  q (r ~q) ....(Identity law)
  
  (q r) (q ~q) ....(Distributive law)
  
  (q r) T ....(Complement law)
  
  q r ....(Identity law)
  
  Simplified switching circuit is:
  
  S1
  
  S2
  
  L
  
  '
  
  1S
  
  '
  
  2S
  
  S1
  
  S3
  
  S2
  
  L
  
  1S
  
  2S
  
  2S
  
  S1
  
  S2
  
  S2
  
  S3
  
  L
  
  • 22
  
  Std. XII : Perfect Maths - I
  
  5. Find the symbolic form of the following
  
  switching circuit, construct its switching table and interpret your result.
  
  Solution:
  
  Let p : The Switch S1 is closed.
  
  q : The Switch S2 is closed.
  
  p : The Switch 1S is closed or the switch S1 is
  
  open.
  
  q : The switch 2S is closed or the switch S2 is
  
  open.
  
  The symbolic form of the given circuit is:
  
  (pq) (p) (q)
  
  Switching table:
  
  p q p q pq (pq) (p) (q)
  
  T T F F T F
  
  T F F T T F
  
  F T T F T F
  
  F F T T F F
  
  In the above truth table, all the entries in the last column are F,
  
  the given circuit represents a contradiction.
  
  Irrespective of whether the switches S1 and S2
  
  are open or closed, the given circuit will always be open (i.e. off).
  
  6. Simplify the given circuit by writing its logical
  
  Solution: Let, p : The switch S1 is closed
  
  q : The switch S2 is closed
  
  p : The switch 1S is closed or the switch S1 is
  
  open.
  
  q : The switch 2S is closed or the switch S2 is
  
  open.
  
  The symbolic form of the given circuit is
  
  [p (~p ~q)] q
  
  [(p ~p) (p ~q)] q
  
  .[Associative and Distributive law]
  
  [F (p ~q)] q .[Complement law]
  
  (p ~q) q .[Identity law]
  
  (p q) (~q q) .[Distributive law]
  
  (p q) F .[Complement law]
  
  F .[Identity law]
  
  Irrespective of the status of the switches, the
  
  current will not flow in the circuit, that is, the circuit will always be open.
  
  Miscellaneous Exercise - 1
  
  1. Which of the following sentences are statements in logic? Justify your answer.
  
  i is a real number.
  
  ii. 5! = 120
  
  iii. Himalaya is an ocean and Ganga is a river.
  
  iv. Please get me a cup of tea.
  
  v. Bring me a notebook.
  
  vi. Alas! We lost the match
  
  vii. cos 2 = cos2 sin2 , for all R.
  
  viii. If x is a real number then x2 0.
  
  Solution:
  
  i. It is a statement.
  
  ii. It is a statement.
  
  iii. It is a statement.
  
  iv. It is an imperative sentence, hence it is not a statement.
  
  v. It is an imperative sentence, hence it is not a statement.
  
  vi. It is an exclamatory sentence, hence it is not a statement.
  
  vii. It is a statement.
  
  viii. It is a statement.
  
  2. Write the truth values of the following statements:
  
  i. The square of any odd number is even or the cube of any even number is even.
  
  ii. 5 is irrational but 3 + 5 is a
  
  complex number. [Oct 14]
  
  iii. n N, such that n + 5 > 10. [Oct 14]
  
  iv. n N, n + 3 > 5.
  
  2S
  
  S1
  
  S2
  
  1S
  
  L
  
  1S
  
  2S
  
  L
  
  S1 S2
  
  • 23
  
  Chapter 01: Mathematical Logic
  
  v. If ABC is a triangle and all its sides
  
  are equal then each angle has
  
  measure 30.
  
  vi. n N, n2 + n is an even number
  
  while n2 n is an odd number.
  
  Solution:
  
  i. Let p: The square of any odd number is even.
  
  q: The cube of any even number is even.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since the truth value of p is F and that of q is T,
  
  truth value of p q is T
  
  ii. Let p: 5 is irrational.
  
  q: 3 + 5 is a complex number.
  
  The symbolic form of the given statement is
  
  p q
  
  Since the truth value of p is T and that of q is F,
  
  truth value of p q is F. [1 Mark]
  
  iii. Consider the statement, n N, n + 5 > 10
  
  Clearly n 6, nN satisfy n + 5 > 10.
  
  its truth value is T. [1 Mark]
  
  iv. Consider the statement, n N, n + 3 > 5
  
  Q n = 1 and n = 2 N do not satisfy n + 3 > 5
  
  truth value of p is F.
  
  v. Let p: ABC is a triangle and all its sides are
  
  equal.
  
  q: Each angle has measure 30.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since the truth value of p is T and that of q is F,
  
  truth value of p q is F
  
  vi. Let p: n N, n2 + n is an even number.
  
  q: nN, n2 n is an odd number.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since, the truth value of p is T and q is F,
  
  truth value of p q is F
  
  3. If A = {4, 5, 7, 9}, determine the truth value of each of the following quantified
  
  statements.
  
  i. x A, such that x + 2 = 7.
  
  ii. x A, x + 3 < 10.
  
  iii. x A, such that x + 5 9.
  
  iv. x A, such that x is even.
  
  v. x A, 2x 17.
  
  Solution:
  
  i. Since x = 5 A, satisfies x + 2 = 7.
  
  the given statement is true.
  
  Its truth value is T.
  
  ii. Since, x = 7, 9 A, do not satisfy x + 3 < 10.
  
  the given statement is false.
  
  Its truth value is F.
  
  iii. Since, x = 4, 5, 7, 9 A, satisfy x + 5 9.
  
  the given statement is true.
  
  Its truth value is T.
  
  iv. Since, x = 4 A, satisfies x is even.
  
  the given statement is true.
  
  Its truth value is T.
  
  v. Since x = 9 A does not satisfy 2x 17.
  
  the given statement is false.
  
  Its truth value is F.
  
  4. Write negations of the following statements
  
  i. Some buildings in this area are
  
  multistoried.
  
  ii. All parents care for their children.
  
  iii. n N, n + 7 > 6.
  
  iv. x A, such that x + 5 > 8.
  
  Solution: i. All buildings in this area are not multistoried.
  
  ii. Some parents do not care for their children.
  
  iii. n N, such that n + 7 6.
  
  iv. x A, x + 5 8.
  
  5. Write the following statements in symbolic form:
  
  i. Ramesh is cruel or strict.
  
  ii. I am brave is necessary and sufficient condition to climb the Mount Everest.
  
  iii. I can travel by train provided I get my ticket reserved.
  
  iv. Sandeep neither likes tea nor coffee but enjoys a soft-drink.
  
  v. ABC is a triangle only if
  
  AB + BC > AC.
  
  vi. Rajesh is studious but does not get good marks.
  
  Solution: i. Let p: Ramesh is cruel,
  
  q: Ramesh is strict.
  
  The symbolic form of the given statement is
  
  p q.
  
  ii. Let p: I am brave.
  
  q: I can climb the Mount Everest.
  
  The symbolic form of the given statement is
  
  p q.
  
  • 24
  
  Std. XII : Perfect Maths - I
  
  iii. Let p: I can travel by train,
  
  q: I get my ticket reserved.
  
  The symbolic form of the given statement is
  
  q p.
  
  iv. Let p: Sandeep likes tea,
  
  q: Sandeep likes coffee.
  
  r: Sandeep enjoys a soft-drink.
  
  The symbolic form of the given statement is
  
  (~p ~q) r.
  
  v. Let p: ABC is a triangle,
  
  q: AB + BC > AC.
  
  The symbolic form of the given statement is
  
  p q.
  
  vi. Let p: Rajesh is studious,
  
  q: Rajesh gets good marks.
  
  The symbolic form of the given statement is
  
  p ~q.
  
  6. If p : The examinations are approaching,
  
  q : Students study hard, give a verbal statement for each of the following:
  
  i. p q ii. p q
  
  iii. p q iv. p q
  
  v. q p
  
  Solution:
  
  i. The examinations are approaching but the students do not study hard.
  
  ii. The examinations are approaching if and only if the students study hard.
  
  iii. If the examinations are not approaching, then
  
  the students study hard.
  
  iv. The examinations are approaching or the
  
  students study hard.
  
  v. If the students do not study hard then the examinations are not approaching.
  
  7. If p: It is raining, q: The weather is humid,
  
  which of the following statements are
  
  logically equivalent? Justify!
  
  i. If it is not raining then the weather is not humid.
  
  ii. It is raining if and only if the weather is humid.
  
  iii. It is not true that it is not raining or the weather is humid.
  
  iv. It is raining but the weather is not
  
  humid.
  
  v. The weather is humid only if it is
  
  raining.
  
  Solution: The symbolic forms of the given statements are:
  
  i. ~p ~q ii. p q
  
  iii. ~(~p q) iv. p ~q
  
  v. q p
  
  Truth table for all the above statements: p q p q (pq) (i) (ii) (iii) (iv) (v)
  
  T T F F T T T F F T
  
  T F F T F T F T T T
  
  F T T F T F F F F F
  
  F F T T T T T F F T
  
  Note: In the above table, the numbers (i), (ii), (iii), (iv) and (v) represent corresponding statements.
  
  In the above table, the columns of statements (i) and (v)
  
  are same.
  
  They are logically equivalent. Similarly, the columns of statements (iii) and (iv) are same.
  
  They are also logically equivalent.
  
  8. Rewrite the following statements without using the conditional form:
  
  i. If prices increase then the wages rise.
  
  ii. If it is cold, we wear woolen clothes.
  
  iii. I can catch cold if I take cold water bath. Solution:
  
  All these statements are of the form p q p q.
  
  The statements without using conditional form will be,
  
  i. Prices do not increase or the wages rise.
  
  ii. It is not cold or we wear woolen clothes. iii. I do not take cold water bath or I catch cold. 9. If p, q, r are statements with truth values T,
  
  F, T respectively, determine the truth
  
  values of the following:
  
  i. q (p r)
  
  ii. (r p) q
  
  iii. (p q) r
  
  iv. (r q) p [July 16]
  
  v. (p q) (q r) Solution: Truth value of p, q and r are T, F and T respectively.
  
  i. q (p ~r)
  
  F (T ~T)
  
  F (T F)
  
  F T
  
  T Hence, the truth value is T.
  
  ii. (~r p) ~q
  
  (~T T) ~F
  
  (F T) T
  
  • 25
  
  Chapter 01: Mathematical Logic
  
  F T
  
  T
  
  Hence, the truth value is T.
  
  iii. (p q) r
  
  (T F) T
  
  F T
  
  T
  
  Hence, the truth value is T.
  
  iv. (r q) p
  
  (T F) T
  
  (T F) F [1 Mark]
  
  F F
  
  T
  
  Hence, the truth value is T [1 Mark]
  
  v. (p q) (q r)
  
  (T F) (F T)
  
  T T
  
  T
  
  Hence, the truth value is T.
  
  10. Change each of the following statement in
  
  the form if then
  
  i. I shall come provided I finish my
  
  work.
  
  ii. Rights follow from performing the
  
  duties sincerely.
  
  iii. x = 1 only if x2 = x.
  
  iv. The sufficient condition for being rich
  
  is to be rational.
  
  v. Getting bonus is necessary condition
  
  for me to purchase a car.
  
  Solution:
  
  i. If I finish my work then I shall come.
  
  ii. If the duties are performed sincerely then the
  
  rights follow.
  
  iii. If x = 1 then x2 = x.
  
  iv. If a man is rational, then he is rich.
  
  v. If I purchase a car then I get bonus.
  
  11. Write negations of the following statements:
  
  i. 6 is an even number or 36 is a perfect
  
  square.
  
  ii. If diagonals of a parallelogram are perpendicular then it is a rhombus.
  
  iii. If 10 > 5 and 5 < 8 then 8 < 7.
  
  iv. A person is rich if and only if he is a
  
  software engineer.
  
  v. Mangoes are delicious but expensive.
  
  vi. It is false that the sky is not blue.
  
  vii. If the weather is fine then my friends
  
  will come and we go for a picnic.
  
  Solution: i. Let, p: 6 is an even number.
  
  q: 36 is a perfect square.
  
  The given statement is of the form p q
  
  Its Negation is p q
  
  i.e. 6 is not an even number and 36 is not a perfect square.
  
  ii. Let, p: Diagonals of a parallelogram are perpendicular.
  
  q : It is a rhombus.
  
  The given statement is of the form p q.
  
  Its negation is p q.
  
  i.e. diagonals of a parallelogram are perpendicular but it is not a rhombus.
  
  iii. Let, p : 10 > 5 and 5 < 8
  
  q : 8 < 7
  
  The given statement is of the form p q
  
  Its negation is p q
  
  i.e, 10 > 5 and 5 < 8 but 8 7.
  
  iv. Let, p: A person is rich.
  
  q: He is a software engineer.
  
  The given statement is of the form p q.
  
  Its negation is (p q) (q p)
  
  i.e., a person is rich and he is not a software
  
  engineer or a person is software engineer and he is not rich.
  
  v. Let, p: Mangoes are delicious.
  
  q: Mangoes are expensive.
  
  The given statement is of the form p q
  
  Its negation is, p q.
  
  i.e., mangoes are not delicious or they are not expensive.
  
  vi. Let, p: Sky is not blue.
  
  The given statement is of the form (p)
  
  Its negation will be, (p) p
  
  i.e., sky is not blue.
  
  vii. Let, p: The weather is fine.
  
  q: My friends are coming.
  
  r: We go for a picnic.
  
  The given statement is of the form p (q r)
  
  Its negation is, p (q r) p (q r)
  
  i.e., the weather is fine but my friends are not
  
  coming or we are not going for a picnic.
  
  12. Construct the truth table for each of the
  
  following statement patterns:
  
  i. p (q p)
  
  ii. (~ p ~ q) [~ (p q)]
  
  • 26
  
  Std. XII : Perfect Maths - I
  
  iii. ~(~p ~q) q
  
  iv. [(p q) r] [~r (p q)]
  
  v. [(p q) (q r)] (p r)
  
  Solution:
  
  i. p (q p)
  
  p q q p p (q p)
  
  T T T T
  
  T F T T
  
  F T F T
  
  F F T T
  
  ii. (~p ~ q) [~ (p q)]
  
  p q ~p ~q ~p ~q p q ~(p q) (~p ~q )
  
  [~(p q)]
  
  T T F F F T F T
  
  T F F T T F T T
  
  F T T F T F T T
  
  F F T T T F T T
  
  iii. ~(~p ~q) q
  
  p q ~p ~q ~p ~q ~(~p ~q) ~(~p ~q) q
  
  T T F F F T T
  
  T F F T F T T
  
  F T T F F T T
  
  F F T T T F F
  
  iv. [(p q) r] [~r (p q)]
  
  p q r p q [(p q)
  
  r] ~r
  
  [~r
  
  (p q)]
  
  [(p q) r]
  
  [~r (p q)]
  
  T T T T T F T T
  
  T T F T T T T T
  
  T F T F T F F F
  
  T F F F F T T F
  
  F T T F T F F F
  
  F T F F F T T F
  
  F F T F T F F F
  
  F F F F F T T F
  
  v. [(p q) (q r)] (p r)
  
  p q r ~p ~p q q r (~p q)
  
  (q r) p r
  
  [(~p q)
  
  (q r)]
  
  ( p r)
  
  T T T F T T T T T
  
  T T F F T F F F T
  
  T F T F F T F T T
  
  T F F F F T F F T
  
  F T T T T T T T T
  
  F T F T T F F T T
  
  F F T T T T T T T
  
  F F F T T T T T T
  
  13. Using truth tables show that following
  
  statement patterns are tautologies.
  
  i. [(p q) q] (p)
  
  ii. (p q) (q p)
  
  iii. [p (q r)] [(p q) r]
  
  Solution: i.
  
  p q ~p ~q (p q) [(p q)
  
  ~q ]
  
  [(p q)
  
  ~q] (~p)
  
  T T F F T F T
  
  T F F T F F T
  
  F T T F T F T
  
  F F T T T T T
  
  In the above truth table, all the entries in the
  
  last column are T.
  
  The given statement pattern is a tautology.
  
  ii.
  
  p q p q q p (p q) (q p)
  
  T T T T T
  
  T F F T T
  
  F T T F T
  
  F F T T T
  
  In the above truth table, all the entries in the
  
  last column are T.
  
  The given statement pattern is a tautology.
  
  iii.
  
  p q r
  
  q
  
  r
  
  [p
  
  (q r)] (p q)
  
  [(p q)
  
  r]
  
  [p (q r)]
  
  [(p q)
  
  r]
  
  T T T T T T T T
  
  T T F F F T F T
  
  T F T T T F T T
  
  T F F T T F T T
  
  F T T T T F T T
  
  F T F F T F T T
  
  F F T T T F T T
  
  F F F T T F T T
  
  In the above truth table, all the entries in the
  
  last column are T.
  
  The given statement pattern is a tautology.
  
  • 27
  
  Chapter 01: Mathematical Logic
  
  14. Using truth tables show that following
  
  i. [(p q) p] (q)
  
  ii. (p q) (p q)
  
  Solution:
  
  i.
  
  p q ~p ~q p q [(p q) ~p] [(p q) ~p]
  
  (~q)
  
  T T F F T F F
  
  T F F T T F F
  
  F T T F T T F
  
  F F T T F F F
  
  In the above truth table, all the entries in the last column are F.
  
  The given statement pattern is a contradiction.
  
  ii.
  
  p q ~p ~q p q (~p ~q) (p q)
  
  (~p ~q)
  
  T T F F T F F
  
  T F F T F T F
  
  F T T F F T F
  
  F F T T F T F
  
  In the above truth table, all the entries in the
  
  last column are F.
  
  The given statement pattern is a contradiction.
  
  15. Find truth values of p and q in the following cases:
  
  i. (p q) is T and (p q) is T.
  
  ii. (p q) is T and (p q) q is F.
  
  iii. (p q) is F and (p q) q is T.
  
  Solution:
  
  i. (p q) is T and (p q) is T.
  
  Consider the following truth table:
  
  If (p q) is T and (p q) is T, then both p and
  
  q have to be true i.e. their truth value must be
  
  T.
  
  ii. (p q) is T and (p q) q is F.
  
  Consider the following truth table:
  
  If (p q) is T and (p q) q is F, then p is
  
  true and q is false i.e., truth value of p is T and
  
  that of q is F.
  
  iii. (p q) is F and (p q) q is T
  
  Consider the following truth table:
  
  If (p q) is F and (p q) q is T, then there
  
  are three possibilities for the truth value of
  
  p and q.
  
  Either, p is T and q is F or
  
  p is F and q is T or
  
  both p and q are F.
  
  16. Determine whether the following statement patterns are tautologies, contradictions or
  
  contingencies.
  
  i. (p q) (p ~q)
  
  ii. (p q) (~p q) (p ~q) (~p ~q)
  
  iii. [p (p q)] q
  
  iv. [(p ~q) (~p q)] r
  
  Solution: i.
  
  p q p q ~q p ~q (p q)
  
  (p ~q)
  
  T T T F F F
  
  T F F T T F
  
  F T T F F F
  
  F F T T F F
  
  p q p q p q
  
  T T T T
  
  T F T F
  
  F T T F
  
  F F F F
  
  p q p q (p q) q
  
  T
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  T
  
  T
  
  F
  
  T
  
  F
  
  T
  
  T
  
  T
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  F
  
  F
  
  T
  
  T
  
  T
  
  T
  
  p q p q (p q) q
  
  • 28
  
  Std. XII : Perfect Maths - I
  
  In the above truth table, all the entries in the
  
  last column are F.
  
  (p q) (p ~q) is a contradiction. ii.
  
  p q ~p ~q p q ~p q p ~q ~p ~q
  
  (p q)
  
  (~p q)
  
  (p ~q)
  
  (~p ~q)
  
  T T F F T F F F T
  
  T F F T F F T F T
  
  F T T F F T F F T
  
  F F T T F F F T T In the above truth table, all the entries in the
  
  last column are T.
  
  (p q) (~p q) (p ~q) (~p ~q) is a
  
  tautology.
  
  iii.
  
  p q p q p (p q) [p (p q)] q
  
  T T T T T
  
  T F F F T
  
  F T T F T
  
  F F T F T
  
  In the above truth table, all the entries in the last column are T.
  
  [p (p q)] q is a tautology.
  
  iv.
  
  p q r ~p ~q p ~q ~p q (p ~q)
  
  (~p q)
  
  [(p ~q)
  
  (~p q)] r
  
  T T T F F T F T T
  
  T T F F F T F T F
  
  T F T F T T F T T
  
  T F F F T T F T F
  
  F T T T F F T T T
  
  F T F T F F T T F
  
  F F T T T T F T T
  
  F F F T T T F T F In the above truth table, the entries in the last
  
  column are a combination of T and F.
  
  [(p ~q) (~p q)] r is a contingency.
  
  17. Using the rules of logic, prove the following logical equivalences.
  
  i. p q ~(p ~q) ~(q ~p)
  
  ii. ~(p q) (~p q) ~p [Mar 16]
  
  iii. ~p q (p q) ~p
  
  Solution:
  
  i. Consider, RHS = (p q) (q p)
  
  [(p q) (q p)]
  
  .[Negation of disjunction]
  
  [(p q)]
  
  .[Negation of double implication]
  
  p q
  
  .[Negation of negation]
  
  = LHS.
  
  p q (p q) (q p)
  
  ii. Consider,
  
  LHS = (p q) (p q)
  
  (p q) (p q)
  
  . (Negation of disjunction) [1 Mark]
  
  p (q q)
  
  .(Distributive law) [1 Mark]
  
  p T ....(Complement law)
  
  p ....(Identity law) [1 Mark]
  
  = RHS.
  
  (p q) (p q) p
  
  iii. Consider, RHS = (p q) p
  
  (p p) (q p)
  
  . [Distributive law]
  
  (p p) (p q)
  
  .[Commutative law]
  
  F (p q)
  
  . [Complement law]
  
  p q . [Identity law]
  
  = LHS
  
  p q (p q) p
  
  18. Using truth tables prove the following logical equivalences:
  
  i. p q (p q) (p q)
  
  ii. (p q) r p (q r)
  
  Solution:
  
  i. Ref Exercise 1.5 Q2 (ii)
  
  ii. Ref Exercise 1.5 Q2 (iii)
  
  19. Write converse, inverse and contrapositive of the following conditional statements:
  
  i. If an angle is a right angle then its
  
  measure is 90.
  
  ii. If two triangles are congruent then their areas are equal. [Mar 15]
  
  iii. If f(2) = 0 then f(x) is divisible by
  
  (x 2).
  
  • 29
  
  Chapter 01: Mathematical Logic
  
  Solution:
  
  i. Let p : An angle is a right angle,
  
  q : Its measure is 90.
  
  The symbolic form of the given statement is
  
  p q.
  
  Converse: q p
  
  i.e. If the measure of an angle is 90, then it is a
  
  right angle.
  
  Inverse: ~p ~q
  
  i.e. If an angle is not a right angle, then its
  
  measure is not 90.
  
  Contrapositive: ~q ~p
  
  i.e. If the measure of an angle is not 90 then it
  
  is not a right angle.
  
  ii. Let p : Two triangles are congruent,
  
  q : Their areas are equal.
  
  The symbolic form of the given statement is
  
  p q.
  
  Converse: q p
  
  i.e. If areas of two triangles are equal then they
  
  are congruent. [1 Mark]
  
  Inverse: ~p ~q
  
  i.e. If two triangles are not congruent then their
  
  areas are not equal.
  
  Contrapositive: ~q ~p
  
  i.e. If areas of two triangles are not equal then they
  
  are not congruent. [1 Mark]
  
  iii. Let p: f(2) = 0,
  
  q: f(x) is divisible by (x 2)
  
  The symbolic form of the given statement is
  
  p q.
  
  Converse: q p
  
  i.e. If f(x) is divisible by (x 2), then f(2) = 0.
  
  Inverse: ~p ~q
  
  i.e. If f(2) 0, then f(x) is not divisible by
  
  (x 2).
  
  Contrapositive: ~q ~p
  
  i.e. If f(x) is not divisible by (x2) then f(2) 0.
  
  20. Without using truth table, prove that
  
  [(p q) p] q is a tautology.
  
  Solution:
  
  [(p q) p] q
  
  [(p ~p) (q ~p)] q .[Distributive law]
  
  [F (q ~p)] q .[Complement law]
  
  (q ~p) q .[Identity law]
  
  ~(q ~p) q [Conditional law]
  
  [(~q ~(~p)] q ....[Negation of conjunction]
  
  (~q p) q .[Negation of negation]
  
  (p ~q) q ....[Commutative law]
  
  p (~q q) ....[Associative law]
  
  p T ....[Complement law]
  
  T ....[Identity law]
  
  Since, the truth value of the given statement pattern is T, therefore, it is a tautology.
  
  21. Consider following statements.
  
  i. If a person is social then he is happy.
  
  ii. If a person is not social then he is not
  
  happy.
  
  iii. If a person is unhappy then he is not
  
  social.
  
  iv. If a person is happy then he is social.
  
  Identify the pairs of statements having same meaning.
  
  Solution:
  
  Let p: A person is social, q: He is happy.
  
  The symbolic forms of the given statements are:
  
  i. p q
  
  ii. ~p ~q
  
  iii. ~q ~p
  
  iv. q p
  
  Statements (i) and (iii) have same meaning.
  
  Since, a statement and its contrapositive are equivalent.
  
  Also, statements (ii) and (iv) have the same meaning. Since, converse and inverse of a
  
  compound statement are also equivalent.
  
  22. Using the rules of logic, write the negations of the following statements:
  
  i. (p q) (q r)
  
  ii. (p q) (p q)
  
  iii. p (q r)
  
  iv. (p q) r
  
  Solution:
  
  i. ~[(p q) (q ~r)]
  
  ~(p q) ~(q ~r)
  
  .[Negation of conjunction]
  
  (~p ~q) (~q ~ (~r)]
  
  .[Negation of disjunction]
  
  (~p ~q) (~q r)
  
  .[Negation of negation]
  
  ( p q) (r q)
  
  ....[Commutative law]
  
  (~p r) ~q
  
  .[Distributive law]
  
  • 30
  
  Std. XII : Perfect Maths - I
  
  ii. [( p q) (p q)]
  
  ( p q) (p q)
  
  .[Negation of disjunction]
  
  [( p) q] [ p ( q)]
  
  .[Negation of conjunction]
  
  (p q) [ p q)
  
  .[Negation of negation]
  
  iii. [p (q r)] p (q r)
  
  .[Negation of conjunction]
  
  p ( q r)
  
  .[Negation of disjunction]
  
  iv. ~[(p q) r]
  
  ~(p q) ~r
  
  ....[Negation of conjunction]
  
  (p ~q) ~r
  
  ....[Negation of implication]
  
  23. Express the given circuits in symbolic form.
  
  i.
  
  ii.
  
  Solution:
  
  i. Let p: the switch S1 is closed.
  
  q: the switch S2 is closed.
  
  ~p: the switch 1S is closed or the switch S1
  
  is open.
  
  ~q: the switch 2S is closed or the switch S2
  
  is open.
  
  The symbolic form of the given circuit is
  
  (p q) (~p) (p ~q) ii. Let p: the switch S1 is closed
  
  q: the switch S2 is closed
  
  r: the switch S3 is closed
  
  The symbolic form of the given circuit is
  
  (p q) (p r)
  
  24. Construct the switching circuits of the
  
  following statements:
  
  i. (p ~q r) [p (~q ~r)]
  
  ii. [(p r) (~q ~r)] (~p ~r)
  
  Solution:
  
  i. Let p: the switch S1 is closed.
  
  q: the switch S2 is closed.
  
  r: the switch S3 is closed.
  
  ~q: the switch 2S is closed or the switch S2
  
  is open.
  
  ~r: the switch 3S is closed or the switch S3
  
  is open.
  
  Consider the given statement,
  
  (p ~q r) [p (~q ~r)]
  
  p ~q r: represents that the switches S1, 2S
  
  and 3S are connected in series.
  
  p (~q ~r): represents that parallel
  
  combination of 2S and
  
  3S is connected in
  
  series with S1.
  
  Therefore, (p ~q r) [p (~q ~ r)] represents that the circuits corresponding to
  
  [(p ~q r) and [p (~ q ~r)] are connected in parallel with each other.
  
  Hence, the switching circuit of given statement is
  
  ii. Let p: the switch S1 is closed
  
  q: the switch S2 is closed
  
  r: the switch S3 is closed
  
  ~p: the switch 1S is closed or the switch S1
  
  is open.
  
  ~q: the switch 2S is closed or the switch S2
  
  is open.
  
  ~r: the switch 3S is closed or the switch S3 is
  
  open.
  
  Consider the given statement,
  
  [(p r) (~q ~r)] (~p ~r)
  
  (p r) (~q ~r): represents that series combination of S1 and S3 and series combination
  
  of 2S and
  
  3S are connected in parallel.
  
  ~p ~r : represents that 1S and
  
  3S are connected
  
  in series.
  
  S1 S2
  
  1S
  
  1S 2S
  
  L
  
  S1 S1
  
  S2
  
  S1
  
  S3
  
  S1
  
  L
  
  S1 S3
  
  S1
  
  3S
  
  '
  
  2S
  
  2S
  
  2S
  
  L
  
  • 31
  
  Chapter 01: Mathematical Logic
  
  Therefore, [(p r) (~q ~ r)] (~p ~ r) represents that the circuits corresponding to
  
  [(p r) (~q ~ r)] and (~p ~r) are connected in series.
  
  Hence, switching circuit of the given statement is
  
  25. Simplify the following circuits so that the new circuit has minimum number of
  
  switches. Also draw the simplified circuit.
  
  i.
  
  ii.
  
  Solution:
  
  i. Refer to Exercise 1.9 Q.3.
  
  ii. Let, p: The switch S1 is closed.
  
  q: The switch S2 closed.
  
  r: The switch S3 is closed.
  
  s: The switch S4 is closed.
  
  t: The switch S5 is closed.
  
  ~r : The switch 3S is closed or the switch S3 is
  
  open.
  
  ~s: The switch 4S is closed or the switch S4 is
  
  open.
  
  ~t: The switch 5S is closed or the switch S5 is
  
  open.
  
  The symbolic form of the given circuit is,
  
  [(p q) (~r ~s ~t) ] [(p q) (r s t)]
  
  [(p q) ~(r s t)] [(p q) (r s t)]
  
  .[De-Morgans law]
  
  (p q) [~(r s t) (r s t)]
  
  .[Distributive law]
  
  (p q) F .[Complement law]
  
  p q .[Identity law]
  
  Hence, the simplified circuit is
  
  26. Check whether the following switching
  
  circuits are logically equivalent. Justify!
  
  A. i.
  
  ii.
  
  B. i.
  
  ii.
  
  Solution:
  
  Let,
  
  p : The switch S1 is closed.
  
  q : The switch S2 is closed.
  
  r : The switch S3 is closed.
  
  A. i. The symbolic form of given circuit is,
  
  p (q r) (p q) (p r) .(i)
  
  .[distributive law]
  
  3S
  
  1S
  
  2S
  
  S3
  
  S1 S2
  
  S2
  
  S3
  
  S1
  
  S2
  
  S1 S2
  
  S3S1
  
  L
  
  S1 S1
  
  2S
  
  3S
  
  S1 S3 S1 S1 1S3S
  
  S1
  
  2S
  
  1S S2
  
  2S
  
  1S
  
  L
  
  L
  
  S1 S2
  
  S4 S5
  
  S1 S2
  
  3S
  
  4S
  
  5S
  
  3S
  
  S1 S2
  
  L
  
  • 32
  
  Std. XII : Perfect Maths - I
  
  ii. The symbolic form of given circuit is,
  
  (p q) (p r) . (ii)
  
  The given circuits are logically equivalent ....[From (i) and (ii)]
  
  B. i. The symbolic form of the given circuit is
  
  (p q) (q r) ....(i)
  
  ii. The symbolic form of the given circuit is,
  
  p (q r) (p q) (p r) .[distributive law] ....(ii)
  
  The given circuits are not logically equivalent ....[From (i) and (ii)]
  
  27. Give alternative arrangement of the
  
  following circuit, so that the new circuit has
  
  minimum switches only.
  
  Solution: Let, p : The switch S1 is closed.
  
  q : The switch S2 is closed. r : The switch S3 is closed.
  
  p: The switch 1S is closed or the switch S1 is
  
  open.
  
  ~q: The switch 2S is closed or the switch S2 is
  
  open.
  
  The symbolic form of given circuit is,
  
  (p q ~p) (~p q r) (p q r)
  
  (p ~q r)
  
  (p ~p q) (~p q r) (p q r)
  
  (p ~q r) .[Commutative law]
  
  (F q) (~p q r) (p q r) (p ~q r) .[Complement law]
  
  F (~p q r) (p q r) (p ~q r) .[Identity law]
  
  (~p q r) (p q r) (p ~q r) .[Identity Law]
  
  [(~p p) (q r)] (p ~q r) .[Distributive law]
  
  [T (q r)] (p ~q r) .[Complement law]
  
  (q r) (p ~q r) .[Identity law]
  
  [q (p ~q)] r .[Distributive law]
  
  [(q p) (q ~q)] r .[Distributive law]
  
  [(q p) T] r .[Complement law]
  
  (q p) r .[Identity law]
  
  (p q) r .[Commutative law]
  
  The switching circuit corresponding to the
  
  given statement is:
  
  28. Draw the simplified circuit of the following
  
  switching circuit.
  
  Solution: Let p : The switch S1 is closed. q : The switch S2 is closed.
  
  ~p : The switch 1S is closed or the switch S1 is open.
  
  ~q : The switch 2S is closed or the switch S2 is open.
  
  The symbolic form of the given circuit is
  
  (~p q) (p ~q) (p q)
  
  (~p q) [p (q ~q)]
  
  .[Commutative and Distributive law]
  
  (~p q) (p T) .[Complement law]
  
  (~p q) T .[Identity law]
  
  p (q T) .[Commutative law]
  
  ~p T .[Identity law]
  
  T .[Identity law]
  
  The symbolic form of the given circuit is a tautology. Hence, the current will always flow
  
  through the circuit irrespective of whether the
  
  switches are open or closed.
  
  L
  
  S1 S2
  
  S3S2
  
  S1 S2 S3
  
  S1 S3
  
  1S
  
  1S
  
  2S
  
  L
  
  S1 S3
  
  S2
  
  L
  
  S1
  
  S1
  
  S2
  
  S2
  
  1S
  
  2S
  
  L
  
  • 33
  
  Chapter 01: Mathematical Logic
  
  29. Represent the following switching circuit in
  
  symbolic form and construct its switching table. Write your conclusion from the
  
  switching table.
  
  Solution: Let p : The switch S1 is closed.
  
  q : The switch S2 is closed. r : The switch S3 is closed.
  
  ~p : The switch 1S is closed or the switch S1 is open
  
  ~q : The switch 2S is closed or the switch S2 is open
  
  ~r : The switch 3S is closed or the switch S3 is
  
  open.
  
  The symbolic form of the given circuit is
  
  (p ~q ~r) [p (q r)]
  
  The switching table corresponding to the given statements is :
  
  p q r ~q ~r p
  
  (~q) (~r) q r
  
  p
  
  (q r)
  
  [p (~q)
  
  (~r)]
  
  [p(qr)]
  
  T T T F F T T T T
  
  T T F F T T F T T
  
  T F T T F T F T T
  
  T F F T T T F T T
  
  F T T F F F T T F
  
  F T F F T T F F F
  
  F F T T F T F F F
  
  F F F T T T F F F
  
  The final column of the above table is equivalent
  
  to the column of p i.e. column corresponding to
  
  switch S1. Hence, the given circuit is equivalent to
  
  the circuit where only switch S1 is present.
  
  Hence, switching circuit is as follows:
  
  Multiple Choice Questions
  
  1. Which of the following is a statement? (A) Stand up!
  
  (B) Will you help me?
  
  (C) Do you like social studies? (D) 27 is a perfect cube.
  
  2. Which of the following is not a statement?
  
  (A) Please do me a favour. (B) 2 is an even integer.
  
  (C) 2 + 1 = 3.
  
  (D) The number 17 is prime.
  
  3. Which of the following is an open statement?
  
  (A) x is a natural number.
  
  (B) Give me a glass of water. (C) Wish you best of luck.
  
  (D) Good morning to all.
  
  4. Which of the following is not a proposition in logic.
  
  (A) 3 is a prime.
  
  (B) 2 is a irrational.
  
  (C) Mathematics is interesting. (D) 5 is an even integer.
  
  5. If p: The sun has set
  
  q: The moon has risen, then the statement The sun has not set or the
  
  moon has not risen in symbolic form is
  
  written as
  
  (A) ~p ~q (B) ~p q
  
  (C) p ~q (D) p ~q
  
  6. Assuming p: She is beautiful, q: She is clever,
  
  the verbal form of p (~q) is (A) She is beautiful but not clever.
  
  (B) She is beautiful and clever.
  
  (C) She is not beautiful and not clever. (D) She is beautiful or not clever.
  
  7. Let p: It is hot and q: It is raining.
  
  The verbal statement for (p ~q) p is (A) If it is hot and not raining, then it is hot.
  
  (B) If it is hot and raining, then it is hot. (C) If it is hot or raining, then it is not hot.
  
  (D) If it is hot and raining, then it is not hot.
  
  8. Using the statements p: Kiran passed the examination,
  
  the statement It is not true that Kiran passes therefore he is sad in symbolic form is
  
  (A) ~p s (B) ~ (p ~ s)
  
  (C) ~p ~ s (D) ~ (p s)
  
  3S
  
  L
  
  S1S1
  
  S2 S3
  
  2S
  
  S1 S1
  
  L
  
  • 34
  
  Std. XII : Perfect Maths - I
  
  9. Assuming p: She is beautiful, q: She is clever,
  
  the verbal form of ~p (~q) is
  
  (A) She is beautiful but not clever.
  
  (B) She is beautiful and clever.
  
  (C) She is not beautiful and not clever.
  
  (D) She is beautiful or not clever.
  
  10. The converse of the statement If it is raining then it is cool is
  
  (A) If it is cool then it is raining.
  
  (B) If it is not cool then it is raining.
  
  (C) If it is not cool then it is not raining.
  
  (D) If it is not raining then it is not cool.
  
  11. If p and q are simple propositions, then p q is true when
  
  (A) p is true and q is false.
  
  (B) p is false and q is true.
  
  (C) p is true and q is true.
  
  (D) p is false q is false.
  
  12. Which of the following is logically equivalent
  
  to ~[~p q]
  
  (A) p ~q (B) ~p q
  
  (C) ~p q (D) ~p ~q
  
  13. The logically equivalent statement of p q is
  
  (A) ~p q (B) q ~p
  
  (C) ~q p (D) ~q ~p
  
  14. The logically equivalent statement of ~p ~q
  
  is
  
  (A) ~p ~q (B) ~(p q)
  
  (C) ~(p q) (D) p q
  
  15. The contrapositive of (p q) r is
  
  (A) ~r ~p ~q (B) ~r (p q)
  
  (C) r (p q) (D) p (q r)
  
  16. Which of the following propositions is true?
  
  (A) p q ~p ~q
  
  (B) ~(p ~q) ~p q
  
  (C) ~(p q) [~ (p q) ~ (q p)]
  
  (D) ~(~p ~q) ~p q
  
  17. When two statements are connected by the connective if and only if then the compound
  
  statement is called
  
  (A) conjunction of the statements.
  
  (B) disjunction of the statements.
  
  (C) biconditional statement.
  
  (D) conditional statement.
  
  18. If p and q be two statements then the
  
  conjunction of the statements, p q is false when
  
  (A) both p and q are true.
  
  (B) either p or q are true
  
  (C) either p or q or both are false.
  
  (D) both p and q are false.
  
  19. The negation of the statement, The question paper is not easy and we shall not pass is
  
  (A) The question paper is not easy or we shall not pass.
  
  (B) The question paper is not easy implies we shall not pass.
  
  (C) The question paper is easy or we shall pass.
  
  (D) We shall pass implies the question paper is not easy.
  
  20. The statement (p q) (~ p ~ q) is
  
  (B) a tautology.
  
  (C) neither a contradiction nor a tautology.
  
  (D) equivalent to p q.
  
  21. The proposition p ~p is a
  
  (B) contingency.
  
  (C) tautology.
  
  22. The proposition p ~ (p q) is a
  
  (A) tautology
  
  (C) contingency
  
  (D) either (A) or (B)
  
  23. The false statement in the following is
  
  (A) p (~p) is a contradiction.
  
  (B) (p q) (~q ~p) is a contradiction.
  
  (C) ~(~p) p is a tautology.
  
  (D) p (~p) is a tautology.
  
  24. Negation of ~(p q) is
  
  (A) ~p ~q
  
  (B) ~p ~q
  
  (C) p ~q
  
  (D) p ~q
  
  25. The dual of ~(pq) [p(q~r)]is,
  
  (A) ~(pq) [p (q~r)]
  
  (B) (pq) [p (q ~ r)]
  
  (C) ~(pq) [p (q r)]
  
  (D) ~(pq) [p (q ~ r)]
  
  • 35
  
  Chapter 01: Mathematical Logic
  
  26. The symbolic form of the following circuit,
  
  where p: switch S1 is closed.and q: switch S2 is closed, is-
  
  (A) (pq) [p(p~q)]
  
  (B) (~pq) [~p (p~q)]
  
  (C) (pq) [~p(p~q)]
  
  (D) (pq) [~p(p~q)]
  
  27. If A = {2, 3, 4, 5, 6}, then which of the
  
  following is not true? [Oct 13]
  
  (A) x A such that x + 3 = 8
  
  (B) x A such that x + 2 < 5
  
  (C) x A such that x + 2 < 9
  
  (D) x A such that x + 6 9
  
  28. If p q = F, p q = F, then the truth value of p and q is : [Oct 15]
  
  (A) T, T (B) T, F (C) F, T (D) F, F
  
  29. The negation of p (q r) is [Mar 16]
  
  (A) p (~ q r)
  
  (B) ~ p (q r)
  
  (C) ~ p (~ q ~ r)
  
  (D) ~ p (q ~ r)
  
  30. Inverse of the statement pattern
  
  (p q) (p q) is [July 16]
  
  (A) (p q) (p q)
  
  (B) (p q) (p q)
  
  (C) ( p q) ( p q)
  
  (D) ( p q) ( p q)
  
  1. (D) 2. (A) 3. (A) 4. (C)
  
  5. (A) 6. (A) 7. (A) 8. (D)
  
  9. (C) 10. (A) 11. (C) 12. (D) 13. (A) 14. (B) 15. (A) 16. (D)
  
  17. (C) 18. (C) 19. (C) 20. (A)
  
  21. (D) 22. (C) 23. (B) 24. (B) 25. (D) 26. (C) 27. (D) 28. (B)
  
  29. (D) 30. (D)
  
  2S
  
  L
  
  S1 S1
  
  S
  
  S1 S1
  
  S1
  
  1S
  
  2S
  
  S1
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  • 1
  
  Chapter 01: Mathematical Logic
  
  Subtopics
  
  1.1 Statement
  
  1.2 Logical Connectives, Compound Statements and Truth Tables
  
  1.3 Statement Pattern and Logical Equivalence Tautology, Contradiction
  
  and Contingency
  
  1.4 Quantifiers and Quantified Statements
  
  1.5 Duality
  
  1.6 Negation of Compound Statement
  
  1.7 Algebra of Statements (Some Standard equivalent Statements)
  
  1.8 Application of Logic to Switching Circuits
  
  Mathematical Logic 01
  
  • 2
  
  Std. XII : Perfect Maths - I
  
  Type of Problems Exercise Q. Nos.
  
  Identify the statements and write
  
  down their Truth Value
  
  1.1 Q.1
  
  Miscellaneous Q.1
  
  Express the statements in Symbolic
  
  Form/Write the statement in
  
  Symbolic Form
  
  1.2 Q.1
  
  1.4 Q.1, 2
  
  Miscellaneous Q.5
  
  Write the Truth values of
  
  Statements
  
  1.2 Q.2
  
  1.4 Q.3, 5
  
  1.6 Q.1
  
  Miscellaneous Q.2, 3, 9
  
  Write the Negation of
  
  Statements/Using the Rules of
  
  Negation write the Negation of
  
  Statements
  
  1.3 Q.1
  
  1.8 Q.1, 2, 4
  
  Miscellaneous Q.4, 11, 22
  
  Write the Verbal statement for the
  
  given Symbolic Statement
  
  1.4 Q.6
  
  Miscellaneous Q.6
  
  Converse, Inverse and
  
  Contrapositive of the statement
  
  1.4 Q.4
  
  Miscellaneous Q.19, 21
  
  Using Quantifiers Convert Open
  
  sentences into True statement 1.6 Q.2
  
  Prepare the Truth Table/Find Truth
  
  Values of p and q for given cases
  
  1.4 Q.7
  
  1.5 Q.1
  
  Miscellaneous Q. 12, 15
  
  Examine the statement Patterns
  
  Contingency)
  
  1.5 Q.3
  
  Miscellaneous Q.13, 14, 16
  
  Using Truth Table, Verify Logical
  
  Equivalence
  
  1.5 Q.2
  
  Miscellaneous Q.7, 18
  
  Write Dual of the statement 1.7 Q.1, 2, 3, 4
  
  Algebra of statements (without
  
  using Truth Table verify the Logical
  
  Equivalence)/Rewrite the statement
  
  without using the conditional form
  
  1.8 Q.3
  
  Miscellaneous Q.8, 17, 20
  
  Change the statements in the form if
  
  then Miscellaneous Q.10
  
  Applications of logic to switching
  
  circuits
  
  1.9 Q.1 to 5
  
  Miscellaneous Q.23 to 29
  
  • 3
  
  Chapter 01: Mathematical Logic
  
  Introduction
  
  Mathematics is an exact science. Every mathematical statement must be precise. Hence,
  
  there has to be proper reasoning in every
  
  mathematical proof.
  
  Proper reasoning involves logic. The study of logic helps in increasing ones ability of systematic and
  
  logical reasoning. It also helps to develop the skills
  
  of understanding various statements and their
  
  validity.
  
  Logic has a wide scale application in circuit designing, computer programming etc. Hence, the
  
  study of logic becomes essential.
  
  Statement and its truth value
  
  There are various means of communication viz., verbal, written etc. Most of the communication
  
  involves the use of language whereby, the ideas are conveyed through sentences.
  
  There are various types of sentences such as: i. Declarative (Assertive)
  
  ii. Imperative (A command or a request)
  
  iii. Exclamatory (Emotions, excitement)
  
  iv. Interrogative (Question)
  
  Statement
  
  A statement is a declarative sentence which is either
  
  true or false but not both simultaneously. Statements are denoted by the letters p, q, r.
  
  For example: i. 3 is an odd number.
  
  ii. 5 is a perfect square.
  
  iii. Sun rises in the east.
  
  iv. x + 3 = 6, when x = 3.
  
  Truth Value
  
  A statement is either True or False. The Truth value of a true statement is defined to be T (TRUE) and that of a false statement is defined to be F
  
  (FALSE).
  
  Note: 0 and 1 can also be used for T and F respectively.
  
  Consider the following statements:
  
  i. There is no prime number between 23 and 29.
  
  ii. The Sun rises in the west.
  
  iii. The square of a real number is negative.
  
  iv. The sum of the angles of a plane triangle is
  
  180.
  
  Here, the truth value of statement i. and iv. is T and that of ii. and iii. is F.
  
  Note:
  
  The sentences like exclamatory, interrogative,
  
  imperative etc., are not considered as statements as the truth value for these statements cannot be
  
  determined.
  
  Open sentence
  
  An open sentence is a sentence whose truth can vary
  
  according to some conditions, which are not stated
  
  in the sentence.
  
  Note:
  
  Open sentence is not considered as statement in logic.
  
  For example:
  
  i. x 5 = 20
  
  This is an open sentence as its truth depends
  
  on value of x (if x = 4, it is true and if x 4, it is false).
  
  ii. Chinese food is very tasty.
  
  This is an open sentence as its truth varies
  
  from individual to individual.
  
  Exercise 1.1
  
  State which of the following sentences are
  
  statements, write down the truth value.
  
  i. The Sun is a star.
  
  ii. May God bless you!
  
  iii. The sum of interior angles of a triangle is
  
  180.
  
  iv. Every real number is a complex number.
  
  v. Why are you upset?
  
  vi. Every quadratic equation has two real
  
  roots.
  
  vii. 9 is a rational number.
  
  viii. x2 3x + 2 = 0, implies that x = 1 or x = 2.
  
  ix. The sum of cube roots of unity is one.
  
  x. Please get me a glass of water.
  
  xi. He is a good person.
  
  xii. Two is the only even prime number.
  
  xiii. sin 2 = 2sin cos for all R.
  
  xiv. What a horrible sight it was!
  
  xv. Do not disturb.
  
  xvi. x2 3x 4 = 0, x = 1.
  
  xvii. Can you speak in French?
  
  xviii. The square of every real number is positive.
  
  xix. It is red in colour.
  
  xx. Every parallelogram is a rhombus.
  
  • 4
  
  Std. XII : Perfect Maths - I
  
  Solution:
  
  i. It is a statement which is true, hence its truth value is T.
  
  ii. It is an exclamatory sentence, hence, it is not a statement.
  
  iii. It is a statement which is true, hence its truth
  
  value is T.
  
  iv. It is a statement which is true, hence its truth
  
  value is T.
  
  v. It is an interrogative sentence, hence it is not a statement.
  
  vi. It is a statement which is false, hence its truth value is F.
  
  vii. It is a statement which is false, hence its truth
  
  value is F.
  
  viii. It is a statement which is false, hence its truth
  
  value is F.
  
  ix. It is a statement which is false, hence its truth value is F.
  
  x. It is an imperative sentence, hence it is not a statement.
  
  xi. It is an open sentence, hence it is not a
  
  statement.
  
  xii. It is a statement which is true, hence its truth
  
  value is T.
  
  xiii. It is a statement which is true, hence its truth value is T.
  
  xiv. It is an exclamatory sentence, hence it is not a statement.
  
  xv. It is an imperative sentence, hence it is not a
  
  statement.
  
  xvi. It is a statement which is true, hence its truth
  
  value is T.
  
  xvii. It is an interrogative sentence, hence, it is not a statement.
  
  xviii. It is a statement which is false, hence its truth value is F. (Since, 0 is a real number and
  
  square of 0 is 0 which is neither positive nor
  
  negative).
  
  xix. It is an open sentence, hence it is not a
  
  statement. (The truth of this sentence depends
  
  upon the reference for the pronoun It.)
  
  xx. It is a statement which is false, hence its truth
  
  value is F.
  
  Logical Connectives, Compound Statements and
  
  Truth Tables
  
  Logical Connectives: The words or group of words such as and, or, if .
  
  then, if and only if, not are used to join or connect
  
  two or more simple sentences. These connecting words are called logical connectives.
  
  Compound Statements:
  
  The new statement that is formed by combining two or more simple statements by using logical
  
  connectives are called compound statements.
  
  Component Statements:
  
  The simple statements that are joined using logical
  
  connectives are called component statements.
  
  For example:
  
  Consider the following simple statements,
  
  i. e is a vowel
  
  ii. b is a consonant
  
  These two component statements can be joined by
  
  using the logical connective or as shown below:
  
  e is a vowel or b is a consonant
  
  The above statement is called compound statement formed by using logical connective or.
  
  Truth Table
  
  A table that shows the relationship between truth values of simple statements and the truth values of
  
  compounds statements formed by using these simple
  
  statements is called truth table.
  
  Note:
  
  The truth value of a compoud statement depends
  
  upon the truth values of its component statements.
  
  Logical Connectives
  
  A. AND [ ] (Conjunction):
  
  If p and q are any two statements connected by the word and, then the resulting
  
  compound statement p and q is called
  
  conjunction of p and q which is written in the
  
  symbolic form as p q.
  
  For example:
  
  p: Today is a pleasant day.
  
  q: I want to go for shopping.
  
  The conjunction of above two statements is
  
  p q i.e. Today is a pleasant day and I want to go for shopping.
  
  A conjunction is true if and only if both p and q are true.
  
  Truth table for conjunction of p and q is as
  
  shown below:
  
  p q p q
  
  T T T T F F
  
  F T F F F F
  
  • 5
  
  Chapter 01: Mathematical Logic
  
  Note:
  
  The words such as but, yet, still, inspite, though, moreover are also used to connect the
  
  simple statements.
  
  These words are generally used by replacing and.
  
  B. OR [ ] (Disjunction):
  
  If p and q are any two statements connected by
  
  the word or, then the resulting compound
  
  statement p or q is called disjunction of p and
  
  q which is written in the symbolic form as
  
  p q.
  
  The word or is used in English language in
  
  two distinct senses, exclusive and inclusive.
  
  For example:
  
  i. Rahul will pass or fail in the exam.
  
  ii. Candidate must be graduate or
  
  In eg. (i), or indicates that only one of the
  
  two possibilities exists but not both which is
  
  called exclusive sense of or. In eg. (ii), or
  
  indicates that first or second or both the
  
  possibilities may exist which is called
  
  inclusive sense of or.
  
  A disjunction is false only when both p and q
  
  are false.
  
  Truth table for disjunction of p and q is as
  
  shown below:
  
  p q p q
  
  T T T
  
  T F T
  
  F T T
  
  F F F Exercise 1.2
  
  1. Express the following statements in
  
  symbolic form:
  
  i. Mango is a fruit but potato is a
  
  vegetable.
  
  ii. Either we play football or go for
  
  cycling.
  
  iii. Milk is white or grass is green.
  
  iv. Inspite of physical disability, Rahul
  
  stood first in the class.
  
  v. Jagdish stays at home while Shrijeet and Shalmali go for a movie.
  
  Solution: i. Let p : Mango is a fruit, q : Potato is a vegetable.
  
  The symbolic form of the given statement is
  
  p q.
  
  ii. Let p : We play football, q : We go for cycling.
  
  The symbolic form of the given statement is
  
  p q.
  
  iii. Let p : Milk is white, q : Grass is green.
  
  The symbolic form of the given statement is
  
  p q.
  
  iv. Let p : Rahul has physical disability,
  
  q : Rahul stood first in the class.
  
  The given statement can be considered as Rahul has physical disability and he stood first in the
  
  class.
  
  The symbolic form of the given statement is
  
  p q.
  
  v. Let p : Jagdish stays at home,
  
  q : Shrijeet and Shalmali go for a movie.
  
  The given statement can be considered as Jagdish stays at home and Shrijeet and Shalmali go for a movie.
  
  The symbolic form of the given statement is
  
  p q.
  
  2. Write the truth values of following statements.
  
  i. 3 is a rational number or 3 + i is a
  
  complex number.
  
  ii. Jupiter is a planet and Mars is a star.
  
  iii. 2 + 3 5 or 2 3 < 5
  
  iv. 2 0 = 2 and 2 + 0 = 2
  
  v. 9 is a perfect square but 11 is a prime number.
  
  vi. Moscow is in Russia or London is in France.
  
  Solution:
  
  i. Let p : 3 is a rational number,
  
  q : 3 + i is a complex number.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since, truth value of p is F and that of q is T.
  
  truth value of p q is T
  
  ii. Let p : Jupiter is a planet,
  
  q : Mars is a star.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since, truth value of p is T and that of q is F.
  
  truth value of p q is F
  
  • 6
  
  Std. XII : Perfect Maths - I
  
  iii. Let p : 2 + 3 5,
  
  q : 2 3 < 5.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since, truth value of both p and q is F.
  
  truth value of p q is F
  
  iv. Let p : 2 0 = 2, q : 2 + 0 = 2.
  
  The symbolic form of the given statement is
  
  p q. Since, truth value of p is F and that of q is T.
  
  truth value of p q is F
  
  v. Let p : 9 is a perfect square,
  
  q : 11 is a prime number.
  
  The symbolic form of the given statement is
  
  p q. Since, truth value of both p and q is T.
  
  truth value of p q is T
  
  vi. Let p : Moscow is in Russia, q : London is in France.
  
  The symbolic form of the given statement is
  
  p q. Since, truth value of p is T and that of q is F.
  
  truth value of p q is T
  
  C. Not [~] (Negation): If p is any statement then negation of p
  
  i.e., not p is denoted by ~p. Negation of any simple statement p can also be formed by
  
  writing It is not true that or It is false that,
  
  before p.
  
  For example: p : Mango is a fruit.
  
  ~p : Mango is not a fruit.
  
  Truth table for negation is as shown below:
  
  p ~p
  
  T F
  
  F T
  
  Note: If a statement is true its negation is false and vice-versa.
  
  Exercise 1.3
  
  Write negations of the following statements:
  
  i. Rome is in Italy. ii. 5 + 5 = 10
  
  iii. 3 is greater than 4.
  
  iv. John is good in river rafting.
  
  v. is an irrational number. vi. The square of a real number is positive.
  
  vii. Zero is not a complex number.
  
  viii. Re (z) | z |.
  
  ix. The sun sets in the East.
  
  x. It is not true that the mangoes are inexpensive.
  
  Solution:
  
  i. Rome is not in Italy.
  
  ii. 5 + 5 10
  
  iii. 3 is not greater than 4.
  
  iv. John is not good in river rafting.
  
  v. is not an irrational number.
  
  vi. The square of a real number is not positive.
  
  vii. Zero is a complex number.
  
  viii. Re (z) > |z|.
  
  ix. The sun does not set in the East.
  
  x. It is true that the mangoes are inexpensive.
  
  D. If.then (Implication, ) (Conditional):
  
  If p and q are any two simple statements, then
  
  the compound statement, if p then q, meaning statement p implies statement q or
  
  statement q is implied by statement p, is
  
  called a conditional statement and is denoted
  
  by p q or p q.
  
  Here p is called the antecedent (hypothesis)
  
  and q is called the consequent (conclusion).
  
  For example:
  
  Let p: I travel by train.
  
  q: My journey will be cheaper.
  
  Here the conditional statement is
  
  p q: If I travel by train then my journey
  
  will be cheaper.
  
  Conditional statement is false if and only if
  
  antecedent is true and consequent is false.
  
  Truth table for conditional is as shown
  
  below:
  
  p q p q
  
  T T T
  
  T F F
  
  F T T
  
  F F T
  
  Note: Equivalent forms of the conditional statement
  
  p q:
  
  a. p is sufficient for q.
  
  b. q is necessary for p.
  
  c. p implies q.
  
  d. p only if q.
  
  e. q follows from p.
  
  • 7
  
  Chapter 01: Mathematical Logic
  
  E. Converse, Inverse and Contrapositive
  
  statements:
  
  If p q is given, then its
  
  converse is q p
  
  inverse is ~p ~q
  
  contrapositive is ~q ~p
  
  For example:
  
  Let p : Smita is intelligent.
  
  q : Smita will join Medical.
  
  i. q p: If Smita joins Medical then she
  
  is intelligent.
  
  ii. ~p ~q: If Smita is not intelligent then
  
  she will not join Medical.
  
  iii. ~q ~p: If Smita does not join Medical
  
  then she is not intelligent.
  
  Consider, the following truth table:
  
  p q pq ~p ~q qp ~q~p ~p ~q
  
  T T T F F T T T
  
  T F F F T T F T
  
  F T T T F F T F
  
  F F T T T T T T
  
  From the above table, we conclude that
  
  i. a conditional statement and its
  
  contrapositive are always equivalent.
  
  ii. converse and inverse of the conditional
  
  statement are always equivalent.
  
  F. If and only if (Double Implication, )
  
  (Biconditional):
  
  If p and q are any two statements, then
  
  p if and only if q or p iff q is called the biconditional statement and is denoted by
  
  p q. Here, both p and q are called implicants.
  
  For example: Let p : price increases
  
  q : demand falls
  
  Here the Biconditional statement is
  
  p q : Price increases if and only if demand
  
  falls.
  
  A biconditional statement is true if and only if
  
  both the implicants have same truth value.
  
  Truth table for biconditional is as shown below:
  
  p q p q
  
  T T T T F F
  
  F T F
  
  F F T
  
  Exercise 1.4
  
  1. Express the following in symbolic form.
  
  i. I like playing but not singing.
  
  ii. Anand neither likes cricket nor tennis.
  
  iii. Rekha and Rama are twins.
  
  iv. It is not true that i is a real number.
  
  v. Either 25 is a perfect square or 41 is divisible by 7.
  
  vi. Rani never works hard yet she gets
  
  good marks.
  
  vii. Eventhough it is not cloudy, it is still
  
  raining.
  
  Solution:
  
  i. Let p: I like playing, q: I like singing,
  
  The symbolic form of the given statement is
  
  p ~q.
  
  ii. Let p: Anand likes cricket, q: Anand likes tennis.
  
  The symbolic form of the given statement is
  
  ~p ~q.
  
  iii. In this statement and is combining two nouns
  
  and not two simple statements.
  
  Hence, it is not used as a connective, so given statement is a simple statement which can be
  
  symbolically expressed as p itself.
  
  iv. Let, p : i is a real number.
  
  The symbolic form of the given statement is ~p.
  
  v. Let p : 25 is a perfect square,
  
  q : 41 is divisible by 7.
  
  The symbolic form of the given statement is
  
  p q.
  
  vi. Let p : Rani works hard, q : Rani gets good
  
  marks.
  
  The symbolic form of the given statement is
  
  ~p q.
  
  vii. Let p : It is cloudy, q : It is still raining.
  
  The symbolic form of the given statement is
  
  ~p q.
  
  2. If p: girls are happy, q: girls are playing, express the following sentences in symbolic
  
  form.
  
  i. Either the girls are happy or they are
  
  not playing.
  
  ii. Girls are unhappy but they are
  
  playing.
  
  iii. It is not true that the girls are not
  
  playing but they are happy.
  
  • 8
  
  Std. XII : Perfect Maths - I
  
  Solution:
  
  i. p ~q ii. ~p q
  
  iii. ~(~q p)
  
  3. Find the truth value of the following statements.
  
  i. 14 is a composite number or 15 is a
  
  prime number.
  
  ii. Neither 21 is a prime number nor it is
  
  divisible by 3.
  
  iii. It is not true that 4+3i is a real
  
  number.
  
  iv. 2 is the only even prime number and 5
  
  divides 26.
  
  v. Either 64 is a perfect square or 46 is a
  
  prime number.
  
  vi. 3 + 5 > 7 if and only if 4 + 6 < 10.
  
  Solution:
  
  i. Let p : 14 is a composite number,
  
  q : 15 is a prime number.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since, truth value of p is T and that of q is F.
  
  truth value of p q is T.
  
  ii. Let p: 21 is a prime number,
  
  q: 21 is divisible by 3.
  
  The symbolic form of the given statement is
  
  ~p ~q.
  
  Since, truth value of p is F and that of q is T
  
  truth value of ~p ~q is F.
  
  iii. Let p: 4 + 3i is a real number.
  
  The symbolic form of the given statement is ~p.
  
  Since, truth value of p is F.
  
  truth value of ~p is T.
  
  iv. Let p: 2 is the only even prime number,
  
  q: 5 divides 26.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since, truth value of p is T and that of q is F
  
  truth value of p q is F.
  
  v. Let p: 64 is a perfect square,
  
  q: 46 is a prime number.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since, truth value of p is T and that of q is F.
  
  truth value of p q is T
  
  vi. Let p: 3 + 5 > 7, q: 4 + 6 < 10
  
  The symbolic form of the given statement is
  
  p q .
  
  Since, truth value of p is T and that of q is F.
  
  truth value of p q is F
  
  4. State the converse, inverse and
  
  contrapositive of the following conditional
  
  statements:
  
  i. If it rains then the match will be
  
  cancelled.
  
  ii. If a function is differentiable then it is continuous.
  
  iii. If surface area decreases then the pressure increases.
  
  iv. If a sequence is bounded then it is
  
  convergent.
  
  Solution:
  
  i. Let p : It rains, q : the match will be cancelled.
  
  The symbolic form of the given statement is
  
  p q.
  
  Converse: q p
  
  i.e., If the match is cancelled then it rains.
  
  Inverse: ~p ~q
  
  i.e., If it does not rain then the match will not be
  
  cancelled.
  
  Contrapositive: ~q ~p
  
  i.e. If the match is not cancelled then it does
  
  not rain.
  
  ii. Let p: A function is differentiable,
  
  q: It is continuous.
  
  The symbolic form of the given statement is
  
  p q.
  
  Converse: q p
  
  i.e. If a function is continuous then it is
  
  differentiable.
  
  Inverse: ~p ~q
  
  i.e. If a function is not differentiable then it
  
  is not continuous.
  
  Contrapositive: ~q ~p
  
  i.e. If a function is not continuous then it is
  
  not differentiable.
  
  iii. Let p: Surface area decreases,
  
  q: The pressure increases.
  
  The symbolic form of the given statement is
  
  p q
  
  Converse: q p
  
  i.e. If the pressure increases then the surface
  
  area decreases.
  
  • 9
  
  Chapter 01: Mathematical Logic
  
  Inverse: ~p ~q
  
  i.e. If the surface area does not decrease
  
  then the pressure does not increase.
  
  Contrapositive: ~q ~p
  
  i.e. If the pressure does not increase then the
  
  surface area does not decrease.
  
  iv. Let p: A sequence is bounded,
  
  q: It is convergent.
  
  The symbolic form of the given statement is
  
  p q
  
  Converse: q p
  
  i.e. If a sequence is convergent then it is
  
  bounded.
  
  Inverse: ~p ~q
  
  i.e. If a sequence is not bounded then it is
  
  not convergent.
  
  Contrapositive: ~q ~p
  
  i.e. If a sequence is not convergent then it is
  
  not bounded.
  
  5. If p and q are true and r and s are false
  
  statements, find the truth value of the
  
  following statements:
  
  i. (p q) r
  
  ii. p (r s)
  
  iii. (p s) (q r)
  
  iv. ~ (p r) (q s)
  
  Solution:
  
  Given that p and q are T and r and s are F.
  
  i. (p q) r
  
  (T T) F
  
  T F
  
  T
  
  truth value of the given statement is true.
  
  ii. p (r s)
  
  T (F F)
  
  T T
  
  T
  
  truth value of the given statement is true.
  
  iii. (p s) (q r)
  
  (T F) (T F)
  
  T F
  
  F
  
  truth value of the given statement is false.
  
  iv. ~(p r) (q s)
  
  (T T) (F F)
  
  (T) F
  
  F F
  
  F
  
  truth value of the given statement is false.
  
  6. If p: It is daytime, q: It is warm
  
  Give the compound statements in verbal
  
  form denoted by
  
  i. p q [Oct 14]
  
  ii. p q
  
  iii. p q
  
  iv. q p [Oct 14]
  
  Solution:
  
  i. It is daytime but it is not warm. [1 Mark]
  
  ii. It is daytime or it is warm.
  
  iii. If it is daytime then it is warm.
  
  iv. It is warm if and only if it is daytime.
  
  [1 Mark]
  
  7. Prepare the truth tables for the following:
  
  i. p q ii. p (p q)
  
  iii. p q
  
  Solution:
  
  i. p q
  
  p q ~p ~p q
  
  T T F F
  
  T F F F
  
  F T T T
  
  F F T F
  
  ii. p (p q)
  
  p q p q p (p q)
  
  T T T T
  
  T F T T
  
  F T T T
  
  F F F T
  
  iii. p q
  
  p q ~p ~p q
  
  T T F F
  
  T F F T
  
  F T T T
  
  F F T F
  
  • 10
  
  Std. XII : Perfect Maths - I
  
  Statement Pattern and Logical Equivalence
  
  A. Statement Pattern
  
  Let, p, q, r, be simple statements. A compound statement obtained from these simple statements and by using one or more
  
  connectives , , ~, , is called a statement pattern.
  
  Following points must be noted while preparing truth tables of the statement
  
  patterns:
  
  i. Parentheses must be introduced
  
  wherever necessary.
  
  For example:
  
  ~ (p q) and ~ p q are not the same.
  
  ii. If a statement pattern consists of n statements and m connectives, then truth table consists of 2n rows and
  
  (m + n) columns.
  
  B. Logical equivalence
  
  Two logical statements are said to be equivalent if and only if the truth values in their respective columns in the joint truth table
  
  are identical.
  
  If S1 and S2 are logically equivalent statement patterns, we write
  
  S1 S2.
  
  For example:
  
  To prove: p q ~(p ~q)
  
  [Mar 08]
  
  p q p q ~q (p ~q) ~(p ~q)
  
  T T T F F T
  
  T F F T T F
  
  F T F F T F
  
  F F F T T F
  
  In the above truth table, all the entries in the
  
  columns of p q and ~(p ~q) are identical.
  
  p q ~(p ~q).
  
  Note:
  
  i. ~(p q) ~p ~q (De-Morgans 1st Law)
  
  [Mar 96]
  
  p q ~p ~q p q ~(p q) ~p ~q
  
  T T F F T F F
  
  T F F T T F F F T T F T F F
  
  F F T T F T T
  
  ii. ~(p q) ~p ~q (De-Morgans 2nd Law)
  
  p q ~p ~q p q ~(p q) ~p ~q
  
  T T F F T F F T F F T F T T
  
  F T T F F T T
  
  F F T T F T T
  
  iii. p q (~p) q
  
  p q ~p p q ~p q
  
  T T F T T
  
  T F F F F F T T T T
  
  F F T T T
  
  iv. p q (p q) (q p)
  
  [Mar 98, Oct 00, 01,04]
  
  p q p q q p p q (p q) (q p)
  
  T T T T T T
  
  T F F T F F
  
  F T T F F F F F T T T T
  
  Tautology
  
  A statement pattern having truth value always T, irrespective of the truth values of its component
  
  statement is called Tautology.
  
  For example, consider (p q) (q p)
  
  p q p q q p (p q) (q p)
  
  T T T T T
  
  T F F F T
  
  F T F F T
  
  F F T T T
  
  In the above truth table, all the entries in the last column are T.
  
  The given statement pattern is a tautology.
  
  A statement pattern having truth value always F, irrespective of the truth values of its component
  
  identical
  
  identical
  
  identical
  
  identical
  
  l
  
  • 11
  
  Chapter 01: Mathematical Logic
  
  For example, consider p ~ p
  
  p ~ p p ~ p
  
  T F F
  
  F T F
  
  In the above truth table, all the entries in the last column
  
  are F.
  
  The given statement pattern is a contradiction.
  
  Contingency
  
  A statement pattern which is neither a tautology nor
  
  For example, consider (p q) ~(p ~ q)
  
  p q p q ~q p ~q ~(p ~q) (p q)
  
  ~(p ~q)
  
  T T T F F T T
  
  T F F T T F F
  
  F T F F T F F
  
  F F T T T F F
  
  In the above truth table, the entries in the last column are a combination of T and F.
  
  The given statement pattern is neither a
  
  tautology nor a contradiction, it is a
  
  contingency.
  
  Exercise 1.5
  
  1. Prepare the truth table of the following statement patterns:
  
  i. [(p q) q] p
  
  ii. (p q) (~p)
  
  iii. (p q) (~p q)
  
  iv. (p r) (q p)
  
  v. (p ~q) (r p)
  
  Solution:
  
  i. [(p q) q] p
  
  p q p q (p q) q [(p q) q] p
  
  T
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  T
  
  T
  
  F
  
  T
  
  F
  
  T
  
  T
  
  F
  
  T
  
  Note:
  
  Here, the statement pattern consists of 2
  
  statements and 3 connectives. Therefore, the truth table has,
  
  Number of rows = 22 = 4
  
  and number of columns = (2+3) = 5.
  
  ii. (p q) (~p)
  
  p q p q ~p (p q) (~p)
  
  T
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  F
  
  F
  
  F
  
  F
  
  T
  
  T
  
  F
  
  T
  
  T
  
  T
  
  iii. (p q) (~p q)
  
  p q p q ~p ~p q (p q)
  
  (~p q)
  
  T
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  T
  
  F
  
  F
  
  T
  
  T
  
  T
  
  F
  
  T
  
  T
  
  T
  
  T
  
  T
  
  T
  
  iv. (p r) (q p)
  
  p q r p r q p (p r)
  
  (q p)
  
  T
  
  T
  
  T
  
  T
  
  F
  
  F
  
  F
  
  F
  
  T
  
  T
  
  F
  
  F
  
  T
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  T
  
  T
  
  F
  
  F
  
  F
  
  F
  
  T
  
  T
  
  T
  
  F
  
  F
  
  F
  
  F
  
  F
  
  F
  
  T
  
  v. (p ~q) (r p)
  
  p q r ~q p ~ q r p (p ~q)
  
  (r p)
  
  T
  
  T
  
  T
  
  T
  
  F
  
  F
  
  F
  
  F
  
  T
  
  T
  
  F
  
  F
  
  T
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  F
  
  F
  
  T
  
  T
  
  F
  
  F
  
  T
  
  T
  
  T
  
  T
  
  T
  
  T
  
  F
  
  F
  
  T
  
  T
  
  T
  
  F
  
  T
  
  F
  
  F
  
  F
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  T
  
  F
  
  F
  
  2. Using truth tables, prove the following logical equivalences:
  
  i. (p q) ~ (p ~ q)
  
  ii. p q (p q) (~p ~q)
  
  [Mar 13; Oct 15]
  
  iii. (p q) r p (q r) [Oct 14]
  
  iv. p (q r) (p q) (p r)
  
  • 12
  
  Std. XII : Perfect Maths - I
  
  Solution:
  
  i.
  
  1 2 3 4 5 6
  
  p q p q ~q p ~q ~ (p ~q)
  
  T
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  F
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  T
  
  T
  
  T
  
  F
  
  F
  
  F
  
  The entries in the columns 3 and 6 are identical.
  
  (p q) ~ (p ~q)
  
  ii.
  
  1 2 3 4 5 6 7 8
  
  p q p q ~p ~q p q ~p ~q (p q)
  
  (~p ~q)
  
  T T T F F T F T
  
  T F F F T F F F
  
  F T F T F F F F
  
  F F T T T F T T
  
  The entries in columns 3 and 8 are identical.
  
  p q (p q) (~p ~q) [1 Mark]
  
  [1 mark each for column 3 and column 8]
  
  iii.
  
  1 2 3 4 5 6 7
  
  p q r p q q r (p q) r p
  
  (q r)
  
  T T T T T T T
  
  T T F T F F F
  
  T F T F T T T
  
  T F F F T T T
  
  F T T F T T T
  
  F T F F F T T
  
  F F T F T T T
  
  F F F F T T T
  
  The entries in the columns 6 and 7 are identical.
  
  (p q) r p (q r) [1 Mark]
  
  [1 mark each for column 6 and column 7]
  
  iv.
  
  1 2 3 4 5 6 7 8
  
  p q r q r p (q r) p q p r (pq) (pr)
  
  T T T T T T T T
  
  T T F F T T T T
  
  T F T F T T T T
  
  T F F F T T T T
  
  F T T T T T T T
  
  F T F F F T F F
  
  F F T F F F T F
  
  F F F F F F F F
  
  The entries in the columns 5 and 8 are identical
  
  p (q r) (p q) (p r)
  
  3. Using truth tables examine whether the
  
  following statement patterns are tautology,
  
  i. (p ~ q) (p q) [Mar 13]
  
  ii. (~p q) (q p)
  
  iii. (p q) (p r)
  
  iv. [(p q) r] [p (q r)]
  
  v. (p q) (p r)
  
  Solution:
  
  i.
  
  1 2 3 4 5 6
  
  p q ~q p ~q p q (p ~ q)
  
  (p q)
  
  T T F F T F
  
  T F T T F F
  
  F T F F T F F F T F T F
  
  In the above truth table, all the entries in the last
  
  column are F.
  
  (p ~q) (p q) is a contradiction. [1 Mark]
  
  [1 mark each for column 5 and column 6]
  
  ii.
  
  p q ~p ~p q q p (~p q)
  
  (q p)
  
  T T F F T F
  
  T F F F T F
  
  F T T T F F
  
  F F T F T F
  
  In the above truth table, all the entries in the last
  
  column are F.
  
  (~p q) (q p) is a contradiction.
  
  • 13
  
  Chapter 01: Mathematical Logic
  
  iii.
  
  p q r p q p r (p q) (p r)
  
  T T T T T T
  
  T T F T F T
  
  T F T F T T
  
  T F F F F F
  
  F T T F F F
  
  F T F F F F
  
  F F T F F F
  
  F F F F F F
  
  In the above truth table, the entries in the last column are a combination of T and F.
  
  (p q) (p r) is a contingency.
  
  iv.
  
  p q r p q q r (p q)
  
  r
  
  p (q r) [(pq)r]
  
  [p(q r)]
  
  T T T T T T T T
  
  T T F T T T T T
  
  T F T T T T T T
  
  T F F T F T T T
  
  F T T T T T T T
  
  F T F T T T T T
  
  F F T F T T T T
  
  F F F F F F F T
  
  In the above truth table, all the entries in the last column are T.
  
  [(p q) r] [p (q r)] is a tautology.
  
  v.
  
  p q r p q p r (p q) (p r)
  
  T T T T T T
  
  T T F T T T
  
  T F T T T T
  
  T F F T T T
  
  F T T T T T
  
  F T F T F F
  
  F F T F T F
  
  F F F F F F
  
  In the above truth table, the entries in the last
  
  column are a combination of T and F.
  
  (p q) (p r) is a contingency.
  
  Quantifiers and Quantified Statements
  
  Quantifiers
  
  Quantifiers are the symbols used to denote a group of words or a phrase. Generally, two types of
  
  quantifiers are used. They are as follows:
  
  i. Universal Quantifier:
  
  The symbol stands for all values of and is known as universal quantifier.
  
  For example, Consider A = {1, 2, 3}
  
  Let p: x A, x < 4
  
  Here, the statement p uses the quantifier for
  
  all().
  
  This statement is true if and only if each and
  
  every element of set A satisfies the condition
  
  x < 4 and is false otherwise.
  
  Here, the given statement is true for all the
  
  elements of set A, as 1, 2, 3 satisfy the
  
  condition, x A, x < 4.
  
  ii. Existential Quantifier:
  
  The symbol stands for there exists and is
  
  known as existential quantifier.
  
  For example, Consider A = {4, 14, 66, 70}.
  
  Let p: x A such that x is an odd number.
  
  Here, the statement p uses the quantifier there
  
  exists ().
  
  This statement is true if atleast one element of
  
  set A satisfies the condition x is an odd
  
  number and is false otherwise.
  
  Here, the given statement is false as none of
  
  the elements of set A satisfy the condition,
  
  x A such that x is an odd number.
  
  Quantified statement
  
  The statement containing quantifiers is known as
  
  quantified statement. Generally, an open sentence
  
  with a quantifier becomes a statement and is called quantified statement.
  
  For example:
  
  Use quantifiers to convert open sentence x + 2 < 4
  
  into a statement.
  
  Solution:
  
  x N such that x + 2 < 4, is a true
  
  statement, since x = 1 N satisfies x + 2 < 4.
  
  Exercise 1.6
  
  1. If A = {3, 4, 6, 8}, determine the truth value
  
  of each of the following:
  
  i. x A, such that x + 4 = 7
  
  ii. x A, x + 4 < 10.
  
  iii. x A, x + 5 13.
  
  iv. x A, such that x is odd.
  
  v. x A, such that (x 3) N
  
  • 14
  
  Std. XII : Perfect Maths - I
  
  Solution:
  
  i. Since x = 3 A, satisfies x + 4 = 7
  
  the given statement is true.
  
  Its truth value is T.
  
  ii. Since x = 6, 8 A, do not satisfy x + 4 < 10,
  
  the given statement is false.
  
  Its truth value is F
  
  iii. Since x = 3, 4, 6 A, do not satisfy x + 5 13,
  
  the given statement is false.
  
  Its truth value is F.
  
  iv. Since x = 3 A, satisfies the given statement,
  
  the given statement is true.
  
  Its truth value is T.
  
  v. Since x = 4, 6, 8 A, satisfy (x 3) N,
  
  the given statement is true.
  
  Its truth value is T. 2. Use quantifiers to convert each of the
  
  following open sentences defined on N, into
  
  a true statement: i. x2 = 25 ii. 2x + 3 < 15
  
  iii. x 3 = 11 iv. x2 + 1 5
  
  v. x2 3x + 2 = 0 Solution:
  
  i. x N, such that x2 = 25.
  
  This is a true statement since x = 5 N satisfies x2 = 25.
  
  ii. x N, such that 2x + 3 < 15. This is a true statement since x = 1, 2, 3, 4, 5
  
  N satisfy 2x + 3 < 15.
  
  iii. x N such that x 3 = 11.
  
  This is a true statement since x = 14 N
  
  satisfies x 3 = 11.
  
  iv. x N, such that x2 + 1 5.
  
  This is a true statement since x = 1, 2 N
  
  satisfy x2 + 1 5.
  
  v. x N such that x2 3x + 2 = 0.
  
  This is a true statement since x = 1, 2 N
  
  satisfy x2 3x + 2 = 0.
  
  Duality
  
  Two compound statements S1 and S2 are said to be duals of each other, if one can be obtained from the
  
  other by interchanging and and vice-versa.
  
  The connectives and are duals of each other. Also, a dual is obtained by replacing t by c and c by
  
  t, where t denotes tautology and c denotes
  
  Remarks:
  
  i. The symbol is not changed while finding
  
  the dual.
  
  ii. Dual of a dual is the statement itself.
  
  iii. The special statements t (tautology) and c
  
  (contradiction) are duals of each other.
  
  iv. T is changed to F and vice-versa.
  
  Principle of Duality:
  
  If a compound statement S1 contains only ~, and
  
  and statement S2 arises from S1 by replacing by
  
  and by , then S1 is a tautology if and only if S2 is a contradiction.
  
  Exercise 1.7
  
  1. Write the duals of the following statements:
  
  i. (p q) r
  
  ii. T (p q)
  
  iii. p [q (p q) r]
  
  Solution:
  
  i. (p q) r
  
  ii. F (p q)
  
  iii. p [~q (p q) ~r]
  
  2. Write the dual statement of each of the
  
  following compound statements:
  
  i. Vijay and Vinay cannot speak Hindi.
  
  ii. Ravi or Avinash went to Chennai.
  
  iii. Madhuri has curly hair and brown eyes. [Mar 14]
  
  Solution:
  
  i. Vijay or Vinay cannot speak Hindi.
  
  ii. Ravi and Avinash went to Chennai.
  
  iii. Madhuri has curly hair or brown eyes. [1 Mark]
  
  3. Write the duals of the following statements:
  
  i. (p q) r p (q r)
  
  ii. p (q r) (p q) (p r)
  
  Solution:
  
  i. (p q) r p (q r)
  
  ii. p (q r) (p q) (p r)
  
  4. Write duals of each of the following statements where t is a tautology and c is a contradiction.
  
  i. p q c ii. ~ p (q c)
  
  iii. (p t) (c ~ q)
  
  Solution:
  
  i. p q t ii. ~ p (q t)
  
  iii. (p c) (t ~ q)
  
  • 15
  
  Chapter 01: Mathematical Logic
  
  Negation of compound statement
  
  i. Negation of conjunction:
  
  Negation of the conjunction of two simple
  
  statements is the disjunction of their negations.
  
  i.e. ~(p q) ~p ~q
  
  ii. Negation of disjunction:
  
  Negation of the disjunction of two simple statements is the conjunction of their
  
  negations.
  
  i.e. ~(p q) ~p ~q
  
  iii. Negation of negation:
  
  The negation of negation of a simple
  
  statement is the statement itself.
  
  i.e. If p is a simple statement then (p) p
  
  iv. Negation of conditional (implication)
  
  statement:
  
  The negation of a conditional statement p q is p but not q.
  
  i.e. ~(p q) p ~q
  
  v. Negation of biconditional (double
  
  implication) statement:
  
  The negation of a biconditional statement
  
  p q is the negation of p q or q p.
  
  i.e. ~(p q) (p ~q) (q ~p)
  
  Note:
  
  Negation of statement pattern involving one or
  
  more simple statements p, q, r, . and one or
  
  more connectives , or is obtained by
  
  replacing by , by . Statements p, q, r
  
  are replaced by their negations p, q, r etc. and vice-versa.
  
  vi. Negation of a quantified statement:
  
  While finding the negations of quantified
  
  statements, the word all is replaced by
  
  some and for every is replaced by there
  
  exists and vice-versa.
  
  Algebra of Statements
  
  Some standard equivalent statements:
  
  a. Idempotent Law:
  
  i. p p p
  
  ii. p p p
  
  b. Commutative Law:
  
  i. p q q p
  
  ii. p q q p
  
  c. Associative Law:
  
  i. (p q) r p (q r) p q r
  
  ii. (p q) r p (q r) p q r
  
  d. Distributive Law:
  
  i. p (q r) (p q) (p r)
  
  ii. p (q r) (p q) (p r)
  
  e. Identity Law:
  
  i. p F p
  
  ii. p F F
  
  iii. p T T
  
  iv. p T p
  
  f. Complement Law:
  
  i. p ~p T
  
  ii. p ~p F
  
  g. Involution Law:
  
  i. ~T F
  
  ii. ~F T
  
  iii. ~(~p) p
  
  h. DeMorgans Law:
  
  i. ~(p q) ~p ~q
  
  ii. ~(p q) ~p ~q
  
  i. Absorption Law:
  
  i. p (p q) p
  
  ii. p (p q) p
  
  j. Conditional Law:
  
  i. p q ~p q
  
  ii. p q (~p q) (~q p)
  
  Exercise 1.8
  
  1. Write the negations of following statements.
  
  i. All equilateral triangles are isosceles.
  
  ii. Some real numbers are not complex numbers.
  
  iii. Every student has paid the fees.
  
  iv. n N, n + 1 > 2.
  
  v. x N, x2 + x is even number.
  
  vi. n N, such that n2 = n.
  
  vii. x R, such that x2 < x.
  
  viii. All students of this college live in the
  
  hostel.
  
  ix. Some continuous functions are
  
  differentiable.
  
  x. Democracy survives if the leaders are
  
  not corrupt.
  
  • 16
  
  Std. XII : Perfect Maths - I
  
  xi. The necessary and sufficient condition
  
  for a person to be successful is to be honest.
  
  unequal roots.
  
  Solution:
  
  i. Some equilateral triangles are not isosceles.
  
  ii. All real numbers are complex numbers.
  
  iii. Some students have not paid the fees.
  
  iv. n N, such that n + 1 2.
  
  v. x N, such that x2 + x is not an even number.
  
  vi. n N, n2 n.
  
  vii. x R, x2 x.
  
  viii. Some students of this college do not live in the
  
  hostel.
  
  ix. All continuous functions are not differentiable.
  
  x. Let, p: The leaders are not corrupt.
  
  q: Democracy survives.
  
  The given statement is of the form, p q
  
  Its negation is of the form p q.
  
  i.e. the leaders are not corrupt, but democracy does not survive.
  
  xi. Let, p: A person is successful.
  
  q: A person is honest.
  
  The given statement is of the form, p q
  
  Its negation is of the form,
  
  (p q) (q p)
  
  i.e. a person is successful, but he is not honest
  
  or a person is honest, but he is not successful.
  
  xii. All quadratic equations have equal roots.
  
  2. Using the rules of negation, write the
  
  negations of the following.
  
  i. p (q r) ii. (~p q) r
  
  iii. (p q) (p q)
  
  Solution:
  
  i. ~[p (q r)]
  
  ~p ~(q r) .(Negation of conjunction)
  
  ~p (q ~r) .(Negation of implication)
  
  ii. ~[(~p q) r]
  
  ~(~p q) ~r ....(Negation of conjunction)
  
  [~(~p) ~q] ~r
  
  .... (Negation of disjunction)
  
  (p ~q) ~r .(Negation of negation)
  
  iii. ~[(~p ~q) (p ~q)]
  
  ~(~p ~q) ~(p ~q)
  
  (Negation of disjunction)
  
  [~(~p) ~ (~q)] [~p ~(~q)]
  
  ....(Negation of conjunction)
  
  (p q) (~p q) .(Negation of negation)
  
  3. Without using truth tables, show that
  
  i. p (q p) p q
  
  ii. (p q) (p q) (q r) q r
  
  Solution:
  
  i. L.H.S. = p (q ~p)
  
  (p q) (p ~p)
  
  (Distributive law)
  
  (p q) F .(Complement law)
  
  p q .(Identity law)
  
  L.H.S. = R.H.S.
  
  p (q p) p q
  
  ii. L.H.S . = (p q) (~p q) (~q r)
  
  [(p ~p) q] (~q r)
  
  (Associative and distributive law)
  
  (T q) (~q r)
  
  ....(Complement law)
  
  q (~q r)
  
  ....(Identity law)
  
  (q ~q) (q r)
  
  ....(Distributive law)
  
  T (q r) ....(Complement law)
  
  q r ....(Identity law)
  
  L.H.S. = R.H.S.
  
  (p q) (p q) (q r) q r
  
  4. Form the negations of the following statements
  
  by giving justification.
  
  i. (p q) (~p r)
  
  ii. (q ~r) (p q)
  
  Solution:
  
  i. ~[(p q) (~p r)]
  
  (p q) ~(~p r)
  
  . (Negation of implication)
  
  (p q) [~(~p) ~r]
  
  . (Negation of disjunction)
  
  (p q) (p ~r)
  
  . (Negation of negation) ii. ~[(q ~r) (p q)]
  
  ~(q ~r) ~(p q)
  
  .(Negation of conjunction)
  
  [~q ~(~r)] (~p ~q)
  
  . (Negation of disjunction)
  
  • 17
  
  Chapter 01: Mathematical Logic
  
  (~q r) (~p ~q)
  
  . (Negation of negation)
  
  (~q r) (~q ~p) .(Commutative law)
  
  ~q (r ~p) .(Distributive law)
  
  Application of Logic to Switching Circuits
  
  The working of an electric switch is similar to a
  
  logical statement which has exactly two outcomes,
  
  namely, T or F. A switch also has two outcomes or
  
  results (current flows and current does not flow)
  
  depending upon the status of the switch i.e. (ON or
  
  OFF). This analogy is very useful in solving
  
  problems of circuit design with the help of logic.
  
  Switch
  
  An electric switch is a two state device used for
  
  turning a current on or off.
  
  As shown in the above figures, if the switch is on
  
  i.e., circuit is closed, current passes through the
  
  circuit and vice-versa.
  
  Consider a simple circuit having a switch S, a
  
  battery and a lamp L. When the switch S is closed (i.e., ON, current is flowing through the
  
  circuit), the lamp glows (is on). Similarly, when the
  
  swtich is open (i.e., OFF, current is not flowing through the circuit), the lamp does not glow (is off).
  
  Thus, if p is a statement the switch is closed and if
  
  l is a statement the lamp glows then p is equivalent
  
  to l i.e p l.
  
  Note:
  
  i. ~p means the switch is open. In this case,
  
  the lamp will not glow and thus p l.
  
  ii. If a switch is ON then its truth value is T or
  
  1 and if the switch is OFF, its truth value is
  
  F or 0.
  
  If there are two switches, then they can be
  
  connected in the following ways:
  
  i. Switches S1 and S2 connected in series:
  
  Let p: the switch S1 is closed
  
  q: the switch S2 is closed
  
  l: the lamp glows
  
  In this case, the lamp glows, if and only if
  
  both the switches are closed.
  
  Thus we have, p q l.
  
  Input Output (switching) table:
  
  p q p q
  
  T (1) T (1) T (1)
  
  T (1) F (0) F (0)
  
  F (0) T (1) F (0)
  
  F (0) F (0) F (0)
  
  ii. Switches S1 and S2 are in parallel
  
  Let p : the switch S1 is closed
  
  q : the switch S2 is closed
  
  l : the lamp glows
  
  In this case, the lamp glows, if at least one of the switches is closed.
  
  Thus, we have p q l,
  
  Input output (switching) table:
  
  p q p q
  
  T (1) T (1) T (1)
  
  T (1) F (0) T (1)
  
  F (0) T (1) T (1)
  
  F (0) F (0) F (0)
  
  The above two networks can be combined
  
  to form a complicated network as shown below:
  
  S
  
  closed (On) switch
  
  S
  
  open (Off ) switch
  
  S1
  
  L
  
  S2
  
  S3
  
  Lamp Battery
  
  S
  
  L
  
  S1
  
  L
  
  S2
  
  S1
  
  L S2
  
  • 18
  
  Std. XII : Perfect Maths - I
  
  Let p : The switch S1 is closed
  
  q : The switch S2 is closed
  
  r : The switch S3 is closed
  
  l : The lamp glows
  
  In this case, the lamp glows, if S1 and S2 both are
  
  closed or if S3 is closed.
  
  Thus we have, (p q) r l.
  
  Note: i. If two or more switches in a circuit are
  
  open or closed simultaneously, then they are denoted by same letter and are called
  
  equivalent switches.
  
  ii. Any two switches in a circuit having opposite states are called
  
  complementary switches.
  
  For example, if S1 and S2 are the two
  
  switches such that when S1 is closed, S2 is open and vice-versa, then the switches
  
  S1 and S2 are called complementary
  
  switches and S2 is denoted as 1S . In
  
  such a situation, one of them is
  
  considered as p and the other as ~p or p.
  
  iii. Two circuits are called equivalent if
  
  output of the two circuits is always same.
  
  iv. A circuit is called simpler if it contains
  
  lesser number of switches.
  
  Example :
  
  Express the following circuit in the symbolic form:
  
  Solution:
  
  Let p : The switch S1 is closed.
  
  q : The switch S2 is closed.
  
  r : The switch S3 is closed.
  
  p : The switch 1S is closed.
  
  r: The switch 3S is closed.
  
  l : Lamp L is on.
  
  The lamp L is on if and only if
  
  i. Switch S1 and Switch S2 are closed.
  
  (Q S1 and S2 are in series.)
  
  or ii. Switch 1S and
  
  3S are closed.
  
  (Q1S and
  
  3S are in series)
  
  and iii. Switch S1 or S2 or S3 are closed.
  
  (Q S1, S2, S3 are in parallel)
  
  Symbolic form of the given circuit is,
  
  [(p q) ( p r)] (p q r) l
  
  Generally l is not written and therefore the
  
  symbolic form is
  
  [(p q) ( p r)] (pq r)
  
  Exercise 1.9
  
  1. Represent the following circuits
  
  symbolically and write the input-output or switching table.
  
  i.
  
  ii.
  
  iii.
  
  Solution:
  
  i. Let p: The switch S1 is closed.
  
  q: The switch S2 is closed.
  
  ~p: The switch 1S is closed or the switch S1 is
  
  open.
  
  ~q: The switch 2S is closed or the switch S2 is
  
  open.
  
  The symbolic form of the given circuit is
  
  (p q) (~p ~q)
  
  S1
  
  S2
  
  S3
  
  L
  
  S1 S2
  
  1S
  
  3S
  
  1S
  
  S1
  
  S2
  
  2S
  
  L
  
  L
  
  1S
  
  2S
  
  S3
  
  S1 S2
  
  S3S1
  
  S2
  
  S2
  
  L
  
  1S
  
  • 19
  
  Chapter 01: Mathematical Logic
  
  Input-output Table:
  
  p q ~p ~q p q ~p ~q (p q)
  
  (~p ~q)
  
  1 1 0 0 1 0 1
  
  1 0 0 1 1 0 1
  
  0 1 1 0 1 0 1
  
  0 0 1 1 0 1 1
  
  Switching Table:
  
  p q ~p ~q p q ~p ~q (p q) (~p ~q)
  
  T T F F T F T
  
  T F F T T F T
  
  F T T F T F T
  
  F F T T F T T
  
  ii. Let p: The switch S1 is closed.
  
  q: The switch S2 is closed.
  
  r: The switch S3 is closed.
  
  ~p: The switch 1S is closed or the switch S1 is
  
  open.
  
  ~q: The switch 2S is closed or the switch S2 is
  
  open.
  
  The symbolic form of the given circuit is
  
  [(p q) (~p ~q)] r.
  
  Let a (p q), b (p q)
  
  Input-output Table: p q r ~p ~q a b a b (a b) r
  
  1 1 1 0 0 1 0 1 1
  
  1 1 0 0 0 1 0 1 0
  
  1 0 1 0 1 0 0 0 0
  
  1 0 0 0 1 0 0 0 0
  
  0 1 1 1 0 0 0 0 0
  
  0 1 0 1 0 0 0 0 0
  
  0 0 1 1 1 0 1 1 1
  
  0 0 0 1 1 0 1 1 0
  
  Switching table:
  
  p q r ~p ~q a b a b (a b) r
  
  T T T F F T F T T
  
  T T F F F T F T F
  
  T F T F T F F F F
  
  T F F F T F F F F
  
  F T T T F F F F F
  
  F T F T F F F F F
  
  F F T T T F T T T
  
  F F F T T F T T F
  
  iii. Let p: The switch S1 is closed
  
  q: The switch S2 is closed
  
  r: The switch S3 is closed
  
  ~p: The switch 1S is closed or the switch S1
  
  is open.
  
  The symbolic form of the given circuit is
  
  (p q) q (r ~p)
  
  Input- output Table:
  
  p q r ~p p q (p q)
  
  q (r ~p)
  
  (p q)
  
  q
  
  (r ~p)
  
  1 1 1 0 1 1 1 1
  
  1 1 0 0 1 1 0 0
  
  1 0 1 0 1 0 1 0
  
  1 0 0 0 1 0 0 0
  
  0 1 1 1 1 1 1 1
  
  0 1 0 1 1 1 1 1
  
  0 0 1 1 0 0 1 0
  
  0 0 0 1 0 0 1 0
  
  Switching table:
  
  p q r ~p p q (p q)
  
  q r ~p
  
  (p q) q
  
  (r ~p)
  
  T T T F T T T T
  
  T T F F T T F F
  
  T F T F T F T F
  
  T F F F T F F F
  
  F T T T T T T T
  
  F T F T T T T T
  
  F F T T F F T F
  
  F F F T F F T F
  
  2. Construct the switching circuits of the following statements.
  
  i. [p (~p q)] [(q r) p]
  
  [Mar 15]
  
  ii. (p q r) [p (q r)]
  
  iii. [(p r) (~q r)] (~p ~r)
  
  Solution:
  
  i. Let p: The switch S1 is closed
  
  q: The switch S2 is closed.
  
  r: The switch S3 is closed.
  
  ~p: The switch 1S is closed or the switch S1 is
  
  open.
  
  ~q: The switch 2S is closed or the switch S2 is
  
  open.
  
  ~r : The switch 3S is closed or the switch S3 is
  
  open. [1 Mark]
  
  • 20
  
  Std. XII : Perfect Maths - I
  
  Consider the given statement,
  
  [p ( p q )] [(q r) p].
  
  p (p q): represents that switch S1 is
  
  connected in parallel with the series combination
  
  of 1S and S2.
  
  (q r ) p : represents that switch 1S is
  
  connected in parallel with the series combination
  
  of 2S and S3.
  
  Therefore, [p (p q )] [(q r) p]
  
  represents that the circuits corresponding to [p
  
  ( p q )] and [(q r) p] are connected in parallel with each other. [1 Mark]
  
  Switching Circuit corresponding to the given
  
  statement is:
  
  ii. Let p : The switch S1 is closed.
  
  q : The switch S2 is closed.
  
  r : The switch S3 is closed.
  
  ~p : The switch 1S is closed or the switch S1
  
  is open.
  
  ~r : The switch 3S is closed or the switch
  
  S3 is open.
  
  Consider the given statement,
  
  (p q r) [ p (q r)]
  
  p q r: represents that switches S1, S2, S3
  
  are connected in series.
  
  p (q r): represents that the series
  
  combination of S2 and 3S is connected in
  
  parallel with 1S .
  
  Therefore, (p q r) [ p (q r) represents that the circuits corresponding to
  
  (p q r) and [ p (q r)] are connected in parallel with each other.
  
  Switching circuit corresponding to the given statement is:
  
  iii. Let p: The switch S1 is closed
  
  q: The switch S2 is closed.
  
  r: The switch S3 is closed.
  
  ~p: The switch 1S is closed or the switch S1 is
  
  open.
  
  ~q: The switch 2S is closed or the switch S2 is
  
  open.
  
  ~r : The switch 3S is closed or the switch S3 is
  
  open. Consider the given statement,
  
  [(p r) (q r)] (p r)
  
  (p r) (q r): represents that the series combination of S1 and S3 and series
  
  combination of 2S and
  
  3S are connected in
  
  parallel with each other.
  
  p r : represents that switches 1S and
  
  3S
  
  are connected in series.
  
  Therefore, [(p r) (q r)] ( p r) represents that the circuits corresponding to
  
  [(p r) (q r)] and (p r) are connected in parallel with each other.
  
  Switching Circuit corresponding to the given statement is:
  
  3. Give an alternative arrangement for the
  
  following circuit, so that the new circuit has two switches only. Also write the switching
  
  table. [July 16]
  
  L
  
  3S
  
  1S
  
  S2
  
  S1 S2 S3
  
  3S
  
  S1 S3
  
  0
  
  2S
  
  1S
  
  L
  
  3S
  
  L
  
  S1
  
  1S
  
  2S
  
  2S
  
  1S S2
  
  L
  
  1S S2
  
  S1
  
  2S
  
  S3
  
  1S
  
  [1 Mark]
  
  • 21
  
  Chapter 01: Mathematical Logic
  
  Solution:
  
  Let p: The switch S1 is closed.
  
  q: The switch S2 is closed.
  
  ~p: The switch 1S is closed or the switch S1 is open.
  
  ~q: The switch 2S is closed or the switch S2 is open.
  
  symbolic form of the given circuit is
  
  (p ~q) (~p q) (~p ~q) [1 Mark]
  
  (pq) [p(qq)] (Associative and Distributive law)
  
  (pq) (pT) .(Complement law)
  
  (pq) p ....(Identity law)
  
  (p ~p) (~q p) ....(Distributive law)
  
  [1 Mark]
  
  (p ~p) (~p q) .(Commutative law)
  
  T(~p ~q) .( Complement law)
  
  pq .(Identity law)
  
  [1 Mark]
  
  The alternative arrangement for given circuit is:
  
  [1 Mark]
  
  Switching Table:
  
  p q ~p ~q p ~q ~p q ~p ~q
  
  (p q)
  
  (p q)
  
  (p q)
  
  T T F F F F F F
  
  T F F T T F F T
  
  F T T F F T F T
  
  F F T T F F T T
  
  4. Find
  
  i. symbolic form
  
  ii. switching table and
  
  iii. draw simplified switching circuit for
  
  the following switching circuit.
  
  Solution:
  
  i. Let p: The switch S1 is closed.
  
  q: The switch S2 is closed.
  
  r: The switch S3 is closed.
  
  ~p: The switch 1S is closed or the switch
  
  S1 is open.
  
  ~q: The switch 2S is closed or the switch
  
  S2 is open.
  
  The symbolic form of the given circuit is
  
  (p q) (~p q) (r ~q)
  
  ii. Switching table:
  
  Let a (p q) (p q) (r q)
  
  p q r pq p q pq rq a
  
  T T T T F F F F T
  
  T T F T F F F F T
  
  T F T F F T F T T
  
  T F F F F T F F F
  
  F T T F T F T F T
  
  F T F F T F T F T
  
  F F T F T T F T T
  
  F F F F T T F F F
  
  iii. For simplified switching circuit,
  
  Consider (p q) (~p q) (r ~q)
  
  [(p ~p) q] (r ~q)
  
  ....(Associative and Distributive law)
  
  (T q) (r ~q)....(Complement law)
  
  q (r ~q) ....(Identity law)
  
  (q r) (q ~q) ....(Distributive law)
  
  (q r) T ....(Complement law)
  
  q r ....(Identity law)
  
  Simplified switching circuit is:
  
  S1
  
  S2
  
  L
  
  '
  
  1S
  
  '
  
  2S
  
  S1
  
  S3
  
  S2
  
  L
  
  1S
  
  2S
  
  2S
  
  S1
  
  S2
  
  S2
  
  S3
  
  L
  
  • 22
  
  Std. XII : Perfect Maths - I
  
  5. Find the symbolic form of the following
  
  switching circuit, construct its switching table and interpret your result.
  
  Solution:
  
  Let p : The Switch S1 is closed.
  
  q : The Switch S2 is closed.
  
  p : The Switch 1S is closed or the switch S1 is
  
  open.
  
  q : The switch 2S is closed or the switch S2 is
  
  open.
  
  The symbolic form of the given circuit is:
  
  (pq) (p) (q)
  
  Switching table:
  
  p q p q pq (pq) (p) (q)
  
  T T F F T F
  
  T F F T T F
  
  F T T F T F
  
  F F T T F F
  
  In the above truth table, all the entries in the last column are F,
  
  the given circuit represents a contradiction.
  
  Irrespective of whether the switches S1 and S2
  
  are open or closed, the given circuit will always be open (i.e. off).
  
  6. Simplify the given circuit by writing its logical
  
  Solution: Let, p : The switch S1 is closed
  
  q : The switch S2 is closed
  
  p : The switch 1S is closed or the switch S1 is
  
  open.
  
  q : The switch 2S is closed or the switch S2 is
  
  open.
  
  The symbolic form of the given circuit is
  
  [p (~p ~q)] q
  
  [(p ~p) (p ~q)] q
  
  .[Associative and Distributive law]
  
  [F (p ~q)] q .[Complement law]
  
  (p ~q) q .[Identity law]
  
  (p q) (~q q) .[Distributive law]
  
  (p q) F .[Complement law]
  
  F .[Identity law]
  
  Irrespective of the status of the switches, the
  
  current will not flow in the circuit, that is, the circuit will always be open.
  
  Miscellaneous Exercise - 1
  
  1. Which of the following sentences are statements in logic? Justify your answer.
  
  i is a real number.
  
  ii. 5! = 120
  
  iii. Himalaya is an ocean and Ganga is a river.
  
  iv. Please get me a cup of tea.
  
  v. Bring me a notebook.
  
  vi. Alas! We lost the match
  
  vii. cos 2 = cos2 sin2 , for all R.
  
  viii. If x is a real number then x2 0.
  
  Solution:
  
  i. It is a statement.
  
  ii. It is a statement.
  
  iii. It is a statement.
  
  iv. It is an imperative sentence, hence it is not a statement.
  
  v. It is an imperative sentence, hence it is not a statement.
  
  vi. It is an exclamatory sentence, hence it is not a statement.
  
  vii. It is a statement.
  
  viii. It is a statement.
  
  2. Write the truth values of the following statements:
  
  i. The square of any odd number is even or the cube of any even number is even.
  
  ii. 5 is irrational but 3 + 5 is a
  
  complex number. [Oct 14]
  
  iii. n N, such that n + 5 > 10. [Oct 14]
  
  iv. n N, n + 3 > 5.
  
  2S
  
  S1
  
  S2
  
  1S
  
  L
  
  1S
  
  2S
  
  L
  
  S1 S2
  
  • 23
  
  Chapter 01: Mathematical Logic
  
  v. If ABC is a triangle and all its sides
  
  are equal then each angle has
  
  measure 30.
  
  vi. n N, n2 + n is an even number
  
  while n2 n is an odd number.
  
  Solution:
  
  i. Let p: The square of any odd number is even.
  
  q: The cube of any even number is even.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since the truth value of p is F and that of q is T,
  
  truth value of p q is T
  
  ii. Let p: 5 is irrational.
  
  q: 3 + 5 is a complex number.
  
  The symbolic form of the given statement is
  
  p q
  
  Since the truth value of p is T and that of q is F,
  
  truth value of p q is F. [1 Mark]
  
  iii. Consider the statement, n N, n + 5 > 10
  
  Clearly n 6, nN satisfy n + 5 > 10.
  
  its truth value is T. [1 Mark]
  
  iv. Consider the statement, n N, n + 3 > 5
  
  Q n = 1 and n = 2 N do not satisfy n + 3 > 5
  
  truth value of p is F.
  
  v. Let p: ABC is a triangle and all its sides are
  
  equal.
  
  q: Each angle has measure 30.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since the truth value of p is T and that of q is F,
  
  truth value of p q is F
  
  vi. Let p: n N, n2 + n is an even number.
  
  q: nN, n2 n is an odd number.
  
  The symbolic form of the given statement is
  
  p q.
  
  Since, the truth value of p is T and q is F,
  
  truth value of p q is F
  
  3. If A = {4, 5, 7, 9}, determine the truth value of each of the following quantified
  
  statements.
  
  i. x A, such that x + 2 = 7.
  
  ii. x A, x + 3 < 10.
  
  iii. x A, such that x + 5 9.
  
  iv. x A, such that x is even.
  
  v. x A, 2x 17.
  
  Solution:
  
  i. Since x = 5 A, satisfies x + 2 = 7.
  
  the given statement is true.
  
  Its truth value is T.
  
  ii. Since, x = 7, 9 A, do not satisfy x + 3 < 10.
  
  the given statement is false.
  
  Its truth value is F.
  
  iii. Since, x = 4, 5, 7, 9 A, satisfy x + 5 9.
  
  the given statement is true.
  
  Its truth value is T.
  
  iv. Since, x = 4 A, satisfies x is even.
  
  the given statement is true.
  
  Its truth value is T.
  
  v. Since x = 9 A does not satisfy 2x 17.
  
  the given statement is false.
  
  Its truth value is F.
  
  4. Write negations of the following statements
  
  i. Some buildings in this area are
  
  multistoried.
  
  ii. All parents care for their children.
  
  iii. n N, n + 7 > 6.
  
  iv. x A, such that x + 5 > 8.
  
  Solution: i. All buildings in this area are not multistoried.
  
  ii. Some parents do not care for their children.
  
  iii. n N, such that n + 7 6.
  
  iv. x A, x + 5 8.
  
  5. Write the following statements in symbolic form:
  
  i. Ramesh is cruel or strict.
  
  ii. I am brave is necessary and sufficient condition to climb the Mount Everest.
  
  iii. I can travel by train provided I get my ticket reserved.
  
  iv. Sandeep neither likes tea nor coffee but enjoys a soft-drink.
  
  v. ABC is a triangle only if
  
  AB + BC > AC.
  
  vi. Rajesh is studious but does not get good marks.
  
  Solution: i. Let p: Ramesh is cruel,
  
  q: Ramesh is strict.
  
  The symbolic form of the given statement is
  
  p q.
  
  ii. Let p: I am brave.
  
  q: I can climb the Mount Everest.
  
  The symbolic form of the given statement is
  
  p q.
  
  • 24
  
  Std. XII : Perfect Maths - I
  
  iii. Let p: I can travel by train,
  
  q: I get my ticket reserved.
  
  The symbolic form of the given statement is
  
  q p.
  
  iv. Let p: Sandeep likes tea,
  
  q: Sandeep likes coffee.
  
  r: Sandeep enjoys a soft-drink.
  
  The symbolic form of the given statement is
  
  (~p ~q) r.
  
  v. Let p: ABC is a triangle,
  
  q: AB + BC > AC.
  
  The symbolic form of the given statement is
  
  p q.
  
  vi. Let p: Rajesh is studious,
  
  q: Rajesh gets good marks.
  
  The symbolic form of the given statement is
  
  p ~q.
  
  6. If p : The examinations are approaching,
  
  q : Students study hard, give a verbal statement for each of the following:
  
  i. p q ii. p q
  
  iii. p q iv. p q
  
  v. q p
  
  Solution:
  
  i. The examinations are approaching but the students do not study hard.
  
  ii. The examinations are approaching if and only if the students study hard.
  
  iii. If the examinations are not approaching, then
  
  the students study hard.
  
  iv. The examinations are approaching or the
  
  students study hard.
  
  v. If the students do not study hard then the examinations are not approaching.
  
  7. If p: It is raining, q: The weather is humid,
  
  which of the following statements are
  
  logically equivalent? Justify!
  
  i. If it is not raining then the weather is not humid.
  
  ii. It is raining if and only if the weather is humid.
  
  iii. It is not true that it is not raining or the weather is humid.
  
  iv. It is raining but the weather is not
  
  humid.
  
  v. The weather is humid only if it is
  
  raining.
  
  Solution: The symbolic forms of the given statements are:
  
  i. ~p ~q ii. p q
  
  iii. ~(~p q) iv. p ~q
  
  v. q p
  
  Truth table for all the above statements: p q p q (pq) (i) (ii) (iii) (iv) (v)
  
  T T F F T T T F F T
  
  T F F T F T F T T T
  
  F T T F T F F F F F
  
  F F T T T T T F F T
  
  Note: In the above table, the numbers (i), (ii), (iii), (iv) and (v) represent corresponding statements.
  
  In the above table, the columns of statements (i) and (v)
  
  are same.
  
  They are logically equivalent. Similarly, the columns of statements (iii) and (iv) are same.
  
  They are also logically equivalent.
  
  8. Rewrite the following statements without using the conditional form:
  
  i. If prices increase then the wages rise.
  
  ii. If it is cold, we wear woolen clothes.
  
  iii. I can catch cold if I take cold water bath. Solution:
  
  All these statements are of the form p q p q.
  
  The statements without using conditional form will be,
  
  i. Prices do not increase or the wages rise.
  
  ii. It is not cold or we wear woolen clothes. iii. I do not take cold water bath or I catch cold. 9. If p, q, r are statements with truth values T,
  
  F, T respectively, determine the truth
  
  values of the following:
  
  i. q (p r)
  
  ii. (r p) q
  
  iii. (p q) r
  
  iv. (r q) p [July 16]
  
  v. (p q) (q r) Solution: Truth value of p, q and r are T, F and T respectively.
  
  i. q (p ~r)
  
  F (T ~T)
  
  F (T F)
  
  F T
  
  T Hence, the truth value is T.
  
  ii. (~r p) ~q
  
  (~T T) ~F
  
  (F T) T
  
  • 25
  
  Chapter 01: Mathematical Logic
  
  F T
  
  T
  
  Hence, the truth value is T.
  
  iii. (p q) r
  
  (T F) T
  
  F T
  
  T
  
  Hence, the truth value is T.
  
  iv. (r q) p
  
  (T F) T
  
  (T F) F [1 Mark]
  
  F F
  
  T
  
  Hence, the truth value is T [1 Mark]
  
  v. (p q) (q r)
  
  (T F) (F T)
  
  T T
  
  T
  
  Hence, the truth value is T.
  
  10. Change each of the following statement in
  
  the form if then
  
  i. I shall come provided I finish my
  
  work.
  
  ii. Rights follow from performing the
  
  duties sincerely.
  
  iii. x = 1 only if x2 = x.
  
  iv. The sufficient condition for being rich
  
  is to be rational.
  
  v. Getting bonus is necessary condition
  
  for me to purchase a car.
  
  Solution:
  
  i. If I finish my work then I shall come.
  
  ii. If the duties are performed sincerely then the
  
  rights follow.
  
  iii. If x = 1 then x2 = x.
  
  iv. If a man is rational, then he is rich.
  
  v. If I purchase a car then I get bonus.
  
  11. Write negations of the following statements:
  
  i. 6 is an even number or 36 is a perfect
  
  square.
  
  ii. If diagonals of a parallelogram are perpendicular then it is a rhombus.
  
  iii. If 10 > 5 and 5 < 8 then 8 < 7.
  
  iv. A person is rich if and only if he is a
  
  software engineer.
  
  v. Mangoes are delicious but expensive.
  
  vi. It is false that the sky is not blue.
  
  vii. If the weather is fine then my friends
  
  will come and we go for a picnic.
  
  Solution: i. Let, p: 6 is an even number.
  
  q: 36 is a perfect square.
  
  The given statement is of the form p q
  
  Its Negation is p q
  
  i.e. 6 is not an even number and 36 is not a perfect square.
  
  ii. Let, p: Diagonals of a parallelogram are perpendicular.
  
  q : It is a rhombus.
  
  The given statement is of the form p q.
  
  Its negation is p q.
  
  i.e. diagonals of a parallelogram are perpendicular but it is not a rhombus.
  
  iii. Let, p : 10 > 5 and 5 < 8
  
  q : 8 < 7
  
  The given statement is of the form p q
  
  Its negation is p q
  
  i.e, 10 > 5 and 5 < 8 but 8 7.
  
  iv. Let, p: A person is rich.
  
  q: He is a software engineer.
  
  The given statement is of the form p q.
  
  Its negation is (p q) (q p)
  
  i.e., a person is rich and he is not a software
  
  engineer or a person is software engineer and he is not rich.
  
  v. Let, p: Mangoes are delicious.
  
  q: Mangoes are expensive.
  
  The given statement is of the form p q
  
  Its negation is, p q.
  
  i.e., mangoes are not delicious or they are not expensive.
  
  vi. Let, p: Sky is not blue.
  
  The given statement is of the form (p)
  
  Its negation will be, (p) p
  
  i.e., sky is not blue.
  
  vii. Let, p: The weather is fine.
  
  q: My friends are coming.
  
  r: We go for a picnic.
  
  The given statement is of the form p (q r)
  
  Its negation is, p (q r) p (q r)
  
  i.e., the weather is fine but my friends are not
  
  coming or we are not going for a picnic.
  
  12. Construct the truth table for each of the
  
  following statement patterns:
  
  i. p (q p)
  
  ii. (~ p ~ q) [~ (p q)]
  
  • 26
  
  Std. XII : Perfect Maths - I
  
  iii. ~(~p ~q) q
  
  iv. [(p q) r] [~r (p q)]
  
  v. [(p q) (q r)] (p r)
  
  Solution:
  
  i. p (q p)
  
  p q q p p (q p)
  
  T T T T
  
  T F T T
  
  F T F T
  
  F F T T
  
  ii. (~p ~ q) [~ (p q)]
  
  p q ~p ~q ~p ~q p q ~(p q) (~p ~q )
  
  [~(p q)]
  
  T T F F F T F T
  
  T F F T T F T T
  
  F T T F T F T T
  
  F F T T T F T T
  
  iii. ~(~p ~q) q
  
  p q ~p ~q ~p ~q ~(~p ~q) ~(~p ~q) q
  
  T T F F F T T
  
  T F F T F T T
  
  F T T F F T T
  
  F F T T T F F
  
  iv. [(p q) r] [~r (p q)]
  
  p q r p q [(p q)
  
  r] ~r
  
  [~r
  
  (p q)]
  
  [(p q) r]
  
  [~r (p q)]
  
  T T T T T F T T
  
  T T F T T T T T
  
  T F T F T F F F
  
  T F F F F T T F
  
  F T T F T F F F
  
  F T F F F T T F
  
  F F T F T F F F
  
  F F F F F T T F
  
  v. [(p q) (q r)] (p r)
  
  p q r ~p ~p q q r (~p q)
  
  (q r) p r
  
  [(~p q)
  
  (q r)]
  
  ( p r)
  
  T T T F T T T T T
  
  T T F F T F F F T
  
  T F T F F T F T T
  
  T F F F F T F F T
  
  F T T T T T T T T
  
  F T F T T F F T T
  
  F F T T T T T T T
  
  F F F T T T T T T
  
  13. Using truth tables show that following
  
  statement patterns are tautologies.
  
  i. [(p q) q] (p)
  
  ii. (p q) (q p)
  
  iii. [p (q r)] [(p q) r]
  
  Solution: i.
  
  p q ~p ~q (p q) [(p q)
  
  ~q ]
  
  [(p q)
  
  ~q] (~p)
  
  T T F F T F T
  
  T F F T F F T
  
  F T T F T F T
  
  F F T T T T T
  
  In the above truth table, all the entries in the
  
  last column are T.
  
  The given statement pattern is a tautology.
  
  ii.
  
  p q p q q p (p q) (q p)
  
  T T T T T
  
  T F F T T
  
  F T T F T
  
  F F T T T
  
  In the above truth table, all the entries in the
  
  last column are T.
  
  The given statement pattern is a tautology.
  
  iii.
  
  p q r
  
  q
  
  r
  
  [p
  
  (q r)] (p q)
  
  [(p q)
  
  r]
  
  [p (q r)]
  
  [(p q)
  
  r]
  
  T T T T T T T T
  
  T T F F F T F T
  
  T F T T T F T T
  
  T F F T T F T T
  
  F T T T T F T T
  
  F T F F T F T T
  
  F F T T T F T T
  
  F F F T T F T T
  
  In the above truth table, all the entries in the
  
  last column are T.
  
  The given statement pattern is a tautology.
  
  • 27
  
  Chapter 01: Mathematical Logic
  
  14. Using truth tables show that following
  
  i. [(p q) p] (q)
  
  ii. (p q) (p q)
  
  Solution:
  
  i.
  
  p q ~p ~q p q [(p q) ~p] [(p q) ~p]
  
  (~q)
  
  T T F F T F F
  
  T F F T T F F
  
  F T T F T T F
  
  F F T T F F F
  
  In the above truth table, all the entries in the last column are F.
  
  The given statement pattern is a contradiction.
  
  ii.
  
  p q ~p ~q p q (~p ~q) (p q)
  
  (~p ~q)
  
  T T F F T F F
  
  T F F T F T F
  
  F T T F F T F
  
  F F T T F T F
  
  In the above truth table, all the entries in the
  
  last column are F.
  
  The given statement pattern is a contradiction.
  
  15. Find truth values of p and q in the following cases:
  
  i. (p q) is T and (p q) is T.
  
  ii. (p q) is T and (p q) q is F.
  
  iii. (p q) is F and (p q) q is T.
  
  Solution:
  
  i. (p q) is T and (p q) is T.
  
  Consider the following truth table:
  
  If (p q) is T and (p q) is T, then both p and
  
  q have to be true i.e. their truth value must be
  
  T.
  
  ii. (p q) is T and (p q) q is F.
  
  Consider the following truth table:
  
  If (p q) is T and (p q) q is F, then p is
  
  true and q is false i.e., truth value of p is T and
  
  that of q is F.
  
  iii. (p q) is F and (p q) q is T
  
  Consider the following truth table:
  
  If (p q) is F and (p q) q is T, then there
  
  are three possibilities for the truth value of
  
  p and q.
  
  Either, p is T and q is F or
  
  p is F and q is T or
  
  both p and q are F.
  
  16. Determine whether the following statement patterns are tautologies, contradictions or
  
  contingencies.
  
  i. (p q) (p ~q)
  
  ii. (p q) (~p q) (p ~q) (~p ~q)
  
  iii. [p (p q)] q
  
  iv. [(p ~q) (~p q)] r
  
  Solution: i.
  
  p q p q ~q p ~q (p q)
  
  (p ~q)
  
  T T T F F F
  
  T F F T T F
  
  F T T F F F
  
  F F T T F F
  
  p q p q p q
  
  T T T T
  
  T F T F
  
  F T T F
  
  F F F F
  
  p q p q (p q) q
  
  T
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  T
  
  T
  
  F
  
  T
  
  F
  
  T
  
  T
  
  T
  
  T
  
  F
  
  F
  
  T
  
  F
  
  T
  
  F
  
  T
  
  F
  
  F
  
  F
  
  T
  
  T
  
  T
  
  T
  
  p q p q (p q) q
  
  • 28
  
  Std. XII : Perfect Maths - I
  
  In the above truth table, all the entries in the
  
  last column are F.
  
  (p q) (p ~q) is a contradiction. ii.
  
  p q ~p ~q p q ~p q p ~q ~p ~q
  
  (p q)
  
  (~p q)
  
  (p ~q)
  
  (~p ~q)
  
  T T F F T F F F T
  
  T F F T F F T F T
  
  F T T F F T F F T
  
  F F T T F F F T T In the above truth table, all the entries in the
  
  last column are T.
  
  (p q) (~p q) (p ~q) (~p ~q) is a
  
  tautology.
  
  iii.
  
  p q p q p (p q) [p (p q)] q
  
  T T T T T
  
  T F F F T
  
  F T T F T
  
  F F T F T
  
  In the above truth table, all the entries in the last column are T.
  
  [p (p q)] q is a tautology.
  
  iv.
  
  p q r ~p ~q p ~q ~p q (p ~q)
  
  (~p q)
  
  [(p ~q)
  
  (~p q)] r
  
  T T T F F T F T T
  
  T T F F F T F T F
  
  T F T F T T F T T
  
  T F F F T T F T F
  
  F T T T F F T T T
  
  F T F T F F T T F
  
  F F T T T T F T T
  
  F F F T T T F T F In the above truth table, the entries in the last
  
  column are a combination of T and F.
  
  [(p ~q) (~p q)] r is a contingency.
  
  17. Using the rules of logic, prove the following logical equivalences.
  
  i. p q ~(p ~q) ~(q ~p)
  
  ii. ~(p q) (~p q) ~p [Mar 16]
  
  iii. ~p q (p q) ~p
  
  Solution:
  
  i. Consider, RHS = (p q) (q p)
  
  [(p q) (q p)]
  
  .[Negation of disjunction]
  
  [(p q)]
  
  .[Negation of double implication]
  
  p q
  
  .[Negation of negation]
  
  = LHS.
  
  p q (p q) (q p)
  
  ii. Consider,
  
  LHS = (p q) (p q)
  
  (p q) (p q)
  
  . (Negation of disjunction) [1 Mark]
  
  p (q q)
  
  .(Distributive law) [1 Mark]
  
  p T ....(Complement law)
  
  p ....(Identity law) [1 Mark]
  
  = RHS.
  
  (p q) (p q) p
  
  iii. Consider, RHS = (p q) p
  
  (p p) (q p)
  
  . [Distributive law]
  
  (p p) (p q)
  
  .[Commutative law]
  
  F (p q)
  
  . [Complement law]
  
  p q . [Identity law]
  
  = LHS
  
  p q (p q) p
  
  18. Using truth tables prove the following logical equivalences:
  
  i. p q (p q) (p q)
  
  ii. (p q) r p (q r)
  
  Solution:
  
  i. Ref Exercise 1.5 Q2 (ii)
  
  ii. Ref Exercise 1.5 Q2 (iii)
  
  19. Write converse, inverse and contrapositive of the following conditional statements:
  
  i. If an angle is a right angle then its
  
  measure is 90.
  
  ii. If two triangles are congruent then their areas are equal. [Mar 15]
  
  iii. If f(2) = 0 then f(x) is divisible by
  
  (x 2).
  
  • 29
  
  Chapter 01: Mathematical Logic
  
  Solution:
  
  i. Let p : An angle is a right angle,
  
  q : Its measure is 90.
  
  The symbolic form of the given statement is
  
  p q.
  
  Converse: q p
  
  i.e. If the measure of an angle is 90, then it is a
  
  right angle.
  
  Inverse: ~p ~q
  
  i.e. If an angle is not a right angle, then its
  
  measure is not 90.
  
  Contrapositive: ~q ~p
  
  i.e. If the measure of an angle is not 90 then it
  
  is not a right angle.
  
  ii. Let p : Two triangles are congruent,
  
  q : Their areas are equal.
  
  The symbolic form of the given statement is
  
  p q.
  
  Converse: q p
  
  i.e. If areas of two triangles are equal then they
  
  are congruent. [1 Mark]
  
  Inverse: ~p ~q
  
  i.e. If two triangles are not congruent then their
  
  areas are not equal.
  
  Contrapositive: ~q ~p
  
  i.e. If areas of two triangles are not equal then they
  
  are not congruent. [1 Mark]
  
  iii. Let p: f(2) = 0,
  
  q: f(x) is divisible by (x 2)
  
  The symbolic form of the given statement is
  
  p q.
  
  Converse: q p
  
  i.e. If f(x) is divisible by (x 2), then f(2) = 0.
  
  Inverse: ~p ~q
  
  i.e. If f(2) 0, then f(x) is not divisible by
  
  (x 2).
  
  Contrapositive: ~q ~p
  
  i.e. If f(x) is not divisible by (x2) then f(2) 0.
  
  20. Without using truth table, prove that
  
  [(p q) p] q is a tautology.
  
  Solution:
  
  [(p q) p] q
  
  [(p ~p) (q ~p)] q .[Distributive law]
  
  [F (q ~p)] q .[Complement law]
  
  (q ~p) q .[Identity law]
  
  ~(q ~p) q [Conditional law]
  
  [(~q ~(~p)] q ....[Negation of conjunction]
  
  (~q p) q .[Negation of negation]
  
  (p ~q) q ....[Commutative law]
  
  p (~q q) ....[Associative law]
  
  p T ....[Complement law]
  
  T ....[Identity law]
  
  Since, the truth value of the given statement pattern is T, therefore, it is a tautology.
  
  21. Consider following statements.
  
  i. If a person is social then he is happy.
  
  ii. If a person is not social then he is not
  
  happy.
  
  iii. If a person is unhappy then he is not
  
  social.
  
  iv. If a person is happy then he is social.
  
  Identify the pairs of statements having same meaning.
  
  Solution:
  
  Let p: A person is social, q: He is happy.
  
  The symbolic forms of the given statements are:
  
  i. p q
  
  ii. ~p ~q
  
  iii. ~q ~p
  
  iv. q p
  
  Statements (i) and (iii) have same meaning.
  
  Since, a statement and its contrapositive are equivalent.
  
  Also, statements (ii) and (iv) have the same meaning. Since, converse and inverse of a
  
  compound statement are also equivalent.
  
  22. Using the rules of logic, write the negations of the following statements:
  
  i. (p q) (q r)
  
  ii. (p q) (p q)
  
  iii. p (q r)
  
  iv. (p q) r
  
  Solution:
  
  i. ~[(p q) (q ~r)]
  
  ~(p q) ~(q ~r)
  
  .[Negation of conjunction]
  
  (~p ~q) (~q ~ (~r)]
  
  .[Negation of disjunction]
  
  (~p ~q) (~q r)
  
  .[Negation of negation]
  
  ( p q) (r q)
  
  ....[Commutative law]
  
  (~p r) ~q
  
  .[Distributive law]
  
  • 30
  
  Std. XII : Perfect Maths - I
  
  ii. [( p q) (p q)]
  
  ( p q) (p q)
  
  .[Negation of disjunction]
  
  [( p) q] [ p ( q)]
  
  .[Negation of conjunction]
  
  (p q) [ p q)
  
  .[Negation of negation]
  
  iii. [p (q r)] p (q r)
  
  .[Negation of conjunction]
  
  p ( q r)
  
  .[Negation of disjunction]
  
  iv. ~[(p q) r]
  
  ~(p q) ~r
  
  ....[Negation of conjunction]
  
  (p ~q) ~r
  
  ....[Negation of implication]
  
  23. Express the given circuits in symbolic form.
  
  i.
  
  ii.
  
  Solution:
  
  i. Let p: the switch S1 is closed.
  
  q: the switch S2 is closed.
  
  ~p: the switch 1S is closed or the switch S1
  
  is open.
  
  ~q: the switch 2S is closed or the switch S2
  
  is open.
  
  The symbolic form of the given circuit is
  
  (p q) (~p) (p ~q) ii. Let p: the switch S1 is closed
  
  q: the switch S2 is closed
  
  r: the switch S3 is closed
  
  The symbolic form of the given circuit is
  
  (p q) (p r)
  
  24. Construct the switching circuits of the
  
  following statements:
  
  i. (p ~q r) [p (~q ~r)]
  
  ii. [(p r) (~q ~r)] (~p ~r)
  
  Solution:
  
  i. Let p: the switch S1 is closed.
  
  q: the switch S2 is closed.
  
  r: the switch S3 is closed.
  
  ~q: the switch 2S is closed or the switch S2
  
  is open.
  
  ~r: the switch 3S is closed or the switch S3
  
  is open.
  
  Consider the given statement,
  
  (p ~q r) [p (~q ~r)]
  
  p ~q r: represents that the switches S1, 2S
  
  and 3S are connected in series.
  
  p (~q ~r): represents that parallel
  
  combination of 2S and
  
  3S is connected in
  
  series with S1.
  
  Therefore, (p ~q r) [p (~q ~ r)] represents that the circuits corresponding to
  
  [(p ~q r) and [p (~ q ~r)] are connected in parallel with each other.
  
  Hence, the switching circuit of given statement is
  
  ii. Let p: the switch S1 is closed
  
  q: the switch S2 is closed
  
  r: the switch S3 is closed
  
  ~p: the switch 1S is closed or the switch S1
  
  is open.
  
  ~q: the switch 2S is closed or the switch S2
  
  is open.
  
  ~r: the switch 3S is closed or the switch S3 is
  
  open.
  
  Consider the given statement,
  
  [(p r) (~q ~r)] (~p ~r)
  
  (p r) (~q ~r): represents that series combination of S1 and S3 and series combination
  
  of 2S and
  
  3S are connected in parallel.
  
  ~p ~r : represents that 1S and
  
  3S are connected
  
  in series.
  
  S1 S2
  
  1S
  
  1S 2S
  
  L
  
  S1 S1
  
  S2
  
  S1
  
  S3
  
  S1
  
  L
  
  S1 S3
  
  S1
  
  3S
  
  '
  
  2S
  
  2S
  
  2S
  
  L
  
  • 31
  
  Chapter 01: Mathematical Logic
  
  Therefore, [(p r) (~q ~ r)] (~p ~ r) represents that the circuits corresponding to
  
  [(p r) (~q ~ r)] and (~p ~r) are connected in series.
  
  Hence, switching circuit of the given statement is
  
  25. Simplify the following circuits so that the new circuit has minimum number of
  
  switches. Also draw the simplified circuit.
  
  i.
  
  ii.
  
  Solution:
  
  i. Refer to Exercise 1.9 Q.3.
  
  ii. Let, p: The switch S1 is closed.
  
  q: The switch S2 closed.
  
  r: The switch S3 is closed.
  
  s: The switch S4 is closed.
  
  t: The switch S5 is closed.
  
  ~r : The switch 3S is closed or the switch S3 is
  
  open.
  
  ~s: The switch 4S is closed or the switch S4 is
  
  open.
  
  ~t: The switch 5S is closed or the switch S5 is
  
  open.
  
  The symbolic form of the given circuit is,
  
  [(p q) (~r ~s ~t) ] [(p q) (r s t)]
  
  [(p q) ~(r s t)] [(p q) (r s t)]
  
  .[De-Morgans law]
  
  (p q) [~(r s t) (r s t)]
  
  .[Distributive law]
  
  (p q) F .[Complement law]
  
  p q .[Identity law]
  
  Hence, the simplified circuit is
  
  26. Check whether the following switching
  
  circuits are logically equivalent. Justify!
  
  A. i.
  
  ii.
  
  B. i.
  
  ii.
  
  Solution:
  
  Let,
  
  p : The switch S1 is closed.
  
  q : The switch S2 is closed.
  
  r : The switch S3 is closed.
  
  A. i. The symbolic form of given circuit is,
  
  p (q r) (p q) (p r) .(i)
  
  .[distributive law]
  
  3S
  
  1S
  
  2S
  
  S3
  
  S1 S2
  
  S2
  
  S3
  
  S1
  
  S2
  
  S1 S2
  
  S3S1
  
  L
  
  S1 S1
  
  2S
  
  3S
  
  S1 S3 S1 S1 1S3S
  
  S1
  
  2S
  
  1S S2
  
  2S
  
  1S
  
  L
  
  L
  
  S1 S2
  
  S4 S5
  
  S1 S2
  
  3S
  
  4S
  
  5S
  
  3S
  
  S1 S2
  
  L
  
  • 32
  
  Std. XII : Perfect Maths - I
  
  ii. The symbolic form of given circuit is,
  
  (p q) (p r) . (ii)
  
  The given circuits are logically equivalent ....[From (i) and (ii)]
  
  B. i. The symbolic form of the given circuit is
  
  (p q) (q r) ....(i)
  
  ii. The symbolic form of the given circuit is,
  
  p (q r) (p q) (p r) .[distributive law] ....(ii)
  
  The given circuits are not logically equivalent ....[From (i) and (ii)]
  
  27. Give alternative arrangement of the
  
  following circuit, so that the new circuit has
  
  minimum switches only.
  
  Solution: Let, p : The switch S1 is closed.
  
  q : The switch S2 is closed. r : The switch S3 is closed.
  
  p: The switch 1S is closed or the switch S1 is
  
  open.
  
  ~q: The switch 2S is closed or the switch S2 is
  
  open.
  
  The symbolic form of given circuit is,
  
  (p q ~p) (~p q r) (p q r)
  
  (p ~q r)
  
  (p ~p q) (~p q r) (p q r)
  
  (p ~q r) .[Commutative law]
  
  (F q) (~p q r) (p q r) (p ~q r) .[Complement law]
  
  F (~p q r) (p q r) (p ~q r) .[Identity law]
  
  (~p q r) (p q r) (p ~q r) .[Identity Law]
  
  [(~p p) (q r)] (p ~q r) .[Distributive law]
  
  [T (q r)] (p ~q r) .[Complement law]
  
  (q r) (p ~q r) .[Identity law]
  
  [q (p ~q)] r .[Distributive law]
  
  [(q p) (q ~q)] r .[Distributive law]
  
  [(q p) T] r .[Complement law]
  
  (q p) r .[Identity law]
  
  (p q) r .[Commutative law]
  
  The switching circuit corresponding to the
  
  given statement is:
  
  28. Draw the simplified circuit of the following
  
  switching circuit.
  
  Solution: Let p : The switch S1 is closed. q : The switch S2 is closed.
  
  ~p : The switch 1S is closed or the switch S1 is open.
  
  ~q : The switch 2S is closed or the switch S2 is open.
  
  The symbolic form of the given circuit is
  
  (~p q) (p ~q) (p q)
  
  (~p q) [p (q ~q)]
  
  .[Commutative and Distributive law]
  
  (~p q) (p T) .[Complement law]
  
  (~p q) T .[Identity law]
  
  p (q T) .[Commutative law]
  
  ~p T .[Identity law]
  
  T .[Identity law]
  
  The symbolic form of the given circuit is a tautology. Hence, the current will always flow
  
  through the circuit irrespective of whether the
  
  switches are open or closed.
  
  L
  
  S1 S2
  
  S3S2
  
  S1 S2 S3
  
  S1 S3
  
  1S
  
  1S
  
  2S
  
  L
  
  S1 S3
  
  S2
  
  L
  
  S1
  
  S1
  
  S2
  
  S2
  
  1S
  
  2S
  
  L
  
  • 33
  
  Chapter 01: Mathematical Logic
  
  29. Represent the following switching circuit in
  
  symbolic form and construct its switching table. Write your conclusion from the
  
  switching table.
  
  Solution: Let p : The switch S1 is closed.
  
  q : The switch S2 is closed. r : The switch S3 is closed.
  
  ~p : The switch 1S is closed or the switch S1 is open
  
  ~q : The switch 2S is closed or the switch S2 is open
  
  ~r : The switch 3S is closed or the switch S3 is
  
  open.
  
  The symbolic form of the given circuit is
  
  (p ~q ~r) [p (q r)]
  
  The switching table corresponding to the given statements is :
  
  p q r ~q ~r p
  
  (~q) (~r) q r
  
  p
  
  (q r)
  
  [p (~q)
  
  (~r)]
  
  [p(qr)]
  
  T T T F F T T T T
  
  T T F F T T F T T
  
  T F T T F T F T T
  
  T F F T T T F T T
  
  F T T F F F T T F
  
  F T F F T T F F F
  
  F F T T F T F F F
  
  F F F T T T F F F
  
  The final column of the above table is equivalent
  
  to the column of p i.e. column corresponding to
  
  switch S1. Hence, the given circuit is equivalent to
  
  the circuit where only switch S1 is present.
  
  Hence, switching circuit is as follows:
  
  Multiple Choice Questions
  
  1. Which of the following is a statement? (A) Stand up!
  
  (B) Will you help me?
  
  (C) Do you like social studies? (D) 27 is a perfect cube.
  
  2. Which of the following is not a statement?
  
  (A) Please do me a favour. (B) 2 is an even integer.
  
  (C) 2 + 1 = 3.
  
  (D) The number 17 is prime.
  
  3. Which of the following is an open statement?
  
  (A) x is a natural number.
  
  (B) Give me a glass of water. (C) Wish you best of luck.
  
  (D) Good morning to all.
  
  4. Which of the following is not a proposition in logic.
  
  (A) 3 is a prime.
  
  (B) 2 is a irrational.
  
  (C) Mathematics is interesting. (D) 5 is an even integer.
  
  5. If p: The sun has set
  
  q: The moon has risen, then the statement The sun has not set or the
  
  moon has not risen in symbolic form is
  
  written as
  
  (A) ~p ~q (B) ~p q
  
  (C) p ~q (D) p ~q
  
  6. Assuming p: She is beautiful, q: She is clever,
  
  the verbal form of p (~q) is (A) She is beautiful but not clever.
  
  (B) She is beautiful and clever.
  
  (C) She is not beautiful and not clever. (D) She is beautiful or not clever.
  
  7. Let p: It is hot and q: It is raining.
  
  The verbal statement for (p ~q) p is (A) If it is hot and not raining, then it is hot.
  
  (B) If it is hot and raining, then it is hot. (C) If it is hot or raining, then it is not hot.
  
  (D) If it is hot and raining, then it is not hot.
  
  8. Using the statements p: Kiran passed the examination,
  
  the statement It is not true that Kiran passes therefore he is sad in symbolic form is
  
  (A) ~p s (B) ~ (p ~ s)
  
  (C) ~p ~ s (D) ~ (p s)
  
  3S
  
  L
  
  S1S1
  
  S2 S3
  
  2S
  
  S1 S1
  
  L
  
  • 34
  
  Std. XII : Perfect Maths - I
  
  9. Assuming p: She is beautiful, q: She is clever,
  
  the verbal form of ~p (~q) is
  
  (A) She is beautiful but not clever.
  
  (B) She is beautiful and clever.
  
  (C) She is not beautiful and not clever.
  
  (D) She is beautiful or not clever.
  
  10. The converse of the statement If it is raining then it is cool is
  
  (A) If it is cool then it is raining.
  
  (B) If it is not cool then it is raining.
  
  (C) If it is not cool then it is not raining.
  
  (D) If it is not raining then it is not cool.
  
  11. If p and q are simple propositions, then p q is true when
  
  (A) p is true and q is false.
  
  (B) p is false and q is true.
  
  (C) p is true and q is true.
  
  (D) p is false q is false.
  
  12. Which of the following is logically equivalent
  
  to ~[~p q]
  
  (A) p ~q (B) ~p q
  
  (C) ~p q (D) ~p ~q
  
  13. The logically equivalent statement of p q is
  
  (A) ~p q (B) q ~p
  
  (C) ~q p (D) ~q ~p
  
  14. The logically equivalent statement of ~p ~q
  
  is
  
  (A) ~p ~q (B) ~(p q)
  
  (C) ~(p q) (D) p q
  
  15. The contrapositive of (p q) r is
  
  (A) ~r ~p ~q (B) ~r (p q)
  
  (C) r (p q) (D) p (q r)
  
  16. Which of the following propositions is true?
  
  (A) p q ~p ~q
  
  (B) ~(p ~q) ~p q
  
  (C) ~(p q) [~ (p q) ~ (q p)]
  
  (D) ~(~p ~q) ~p q
  
  17. When two statements are connected by the connective if and only if then the compound
  
  statement is called
  
  (A) conjunction of the statements.
  
  (B) disjunction of the statements.
  
  (C) biconditional statement.
  
  (D) conditional statement.
  
  18. If p and q be two statements then the
  
  conjunction of the statements, p q is false when
  
  (A) both p and q are true.
  
  (B) either p or q are true
  
  (C) either p or q or both are false.
  
  (D) both p and q are false.
  
  19. The negation of the statement, The question paper is not easy and we shall not pass is
  
  (A) The question paper is not easy or we shall not pass.
  
  (B) The question paper is not easy implies we shall not pass.
  
  (C) The question paper is easy or we shall pass.
  
  (D) We shall pass implies the question paper is not easy.
  
  20. The statement (p q) (~ p ~ q) is
  
  (B) a tautology.
  
  (C) neither a contradiction nor a tautology.
  
  (D) equivalent to p q.
  
  21. The proposition p ~p is a
  
  (B) contingency.
  
  (C) tautology.
  
  22. The proposition p ~ (p q) is a
  
  (A) tautology
  
  (C) contingency
  
  (D) either (A) or (B)
  
  23. The false statement in the following is
  
  (A) p (~p) is a contradiction.
  
  (B) (p q) (~q ~p) is a contradiction.
  
  (C) ~(~p) p is a tautology.
  
  (D) p (~p) is a tautology.
  
  24. Negation of ~(p q) is
  
  (A) ~p ~q
  
  (B) ~p ~q
  
  (C) p ~q
  
  (D) p ~q
  
  25. The dual of ~(pq) [p(q~r)]is,
  
  (A) ~(pq) [p (q~r)]
  
  (B) (pq) [p (q ~ r)]
  
  (C) ~(pq) [p (q r)]
  
  (D) ~(pq) [p (q ~ r)]
  
  • 35
  
  Chapter 01: Mathematical Logic
  
  26. The symbolic form of the following circuit,
  
  where p: switch S1 is closed.and q: switch S2 is closed, is-
  
  (A) (pq) [p(p~q)]
  
  (B) (~pq) [~p (p~q)]
  
  (C) (pq) [~p(p~q)]
  
  (D) (pq) [~p(p~q)]
  
  27. If A = {2, 3, 4, 5, 6}, then which of the
  
  following is not true? [Oct 13]
  
  (A) x A such that x + 3 = 8
  
  (B) x A such that x + 2 < 5
  
  (C) x A such that x + 2 < 9
  
  (D) x A such that x + 6 9
  
  28. If p q = F, p q = F, then the truth value of p and q is : [Oct 15]
  
  (A) T, T (B) T, F (C) F, T (D) F, F
  
  29. The negation of p (q r) is [Mar 16]
  
  (A) p (~ q r)
  
  (B) ~ p (q r)
  
  (C) ~ p (~ q ~ r)
  
  (D) ~ p (q ~ r)
  
  30. Inverse of the statement pattern
  
  (p q) (p q) is [July 16]
  
  (A) (p q) (p q)
  
  (B) (p q) (p q)
  
  (C) ( p q) ( p q)
  
  (D) ( p q) ( p q)
  
  1. (D) 2. (A) 3. (A) 4. (C)
  
  5. (A) 6. (A) 7. (A) 8. (D)
  
  9. (C) 10. (A) 11. (C) 12. (D) 13. (A) 14. (B) 15. (A) 16. (D)
  
  17. (C) 18. (C) 19. (C) 20. (A)
  
  21. (D) 22. (C) 23. (B) 24. (B) 25. (D) 26. (C) 27. (D) 28. (B)
  
  29. (D) 30. (D)
  
  2S
  
  L
  
  S1 S1
  
  S
  
  S1 S1
  
  S1
  
  1S
  
  2S
  
  S1
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: ### Document Evaluation
  
  The provided text is a detailed chapter on **Mathematical Logic** from a mathematics textbook. The chapter covers various topics such as statements, logical connectives, truth tables, quantifiers, and applications to switching circuits. The text includes definitions, examples, exercises, and solutions to these exercises. It also contains multiple-choice questions (MCQs) at the end of the chapter.
  
  ### Key Sections:
  1. **Introduction to Mathematical Logic**:
     - Definition of statements and truth values.
     - Types of sentences (declarative, imperative, exclamatory, interrogative).
  
  2. **Logical Connectives**:
     - AND, OR, NOT, IF-THEN, IF AND ONLY IF.
     - Truth tables for these connectives.
  
  3. **Quantifiers**:
     - Universal quantifier (for all) and existential quantifier (there exists).
     - Quantified statements and their truth values.
  
  4. **Duality**:
     - Definition and examples of dual statements.
  
  5. **Negation of Compound Statements**:
     - Rules for negating conjunctions, disjunctions, implications, and biconditionals.
  
  6. **Algebra of Statements**:
     - Standard equivalent statements and their properties (idemp