Consider two propositions generated bypandq: ¬(p∧q)and¬p∨ ¬q. At first glance, they are different propositions. In form, they are different, but they have the same meaning. One way to see this is to substitute actual propositions forpandq; such asp: I’ve been to Toronto; andq: I’ve been to Chicago.
Then¬(p∧q)translates to “I haven’t been to both Toronto and Chicago,” while ¬p∨ ¬q is “I haven’t been to Toronto or I haven’t been to Chicago.” Determine the truth values of these propositions. Naturally, they will be true for some people and false for others. What is important is that no matter what truth values they have, ¬(p∧q) and ¬p∨ ¬q will have the same truth value. The easiest way to see this is by examining the truth tables of these propositions. p q ¬(p∧q) ¬p∨ ¬q 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0 0
Table 3.3.1: Truth Tables for¬(p∧q)and¬p∨ ¬q
In all four cases, ¬(p∧q)and¬p∨ ¬qhave the same truth value. Further- more, when the biconditional operator is applied to them, the result is a value of true in all cases. A proposition such as this is called a tautology.
3.3.1 Tautologies and Contradictions
Definition 3.3.2 Tautology. An expression involving logical variables that is true in all cases is a tautology. The number 1 is used to symbolize a tautology. ♢ Example 3.3.3 Some Tautologies. All of the following are tautologies because their truth tables consist of a column of 1’s.
(a) (¬(p∧q))↔(¬p∨ ¬q). (b) p∨ ¬p (c) (p∧q)→p (d) q→(p∨q) (e) (p∨q)↔(q∨p) □ Definition 3.3.4 Contradiction. An expression involving logical variables that is false for all cases is called a contradiction. The number 0 is used to
CHAPTER 3. LOGIC 48 Example 3.3.5 Some Contradictions. p∧ ¬pand(p∨q)∧(¬p)∧(¬q)are
Definition 3.3.6 Equivalence. LetSbe a set of propositions and letrand sbe propositions generated byS. rands are equivalent if and only ifr↔s is a tautology. The equivalence ofrandsis denotedr ⇐⇒ s. ♢ Equivalence is to logic as equality is to algebra. Just as there are many ways of writing an algebraic expression, the same logical meaning can be expressed in many different ways.
Example 3.3.7 Some Equivalences. The following are all equivalences:
(a) (p∧q)∨(¬p∧q) ⇐⇒ q. (b) p→q ⇐⇒ ¬q→ ¬p
(c) p∨q ⇐⇒ q∨p.
□ All tautologies are equivalent to one another.
Example 3.3.8 An equivalence to 1. p∨ ¬p ⇐⇒ 1. □ All contradictions are equivalent to one another.
Example 3.3.9 An equivalence to 0. p∧ ¬p ⇐⇒ 0. □
Consider the two propositions:
x: The money is behind Door A; and y: The money is behind Door A or Door B.
Imagine that you were told that there is a large sum of money behind one of two doors marked A and B, and that one of the two propositionsx andy is true and the other is false. Which door would you choose? All that you need to realize is that ifxis true, thenywill also be true. Since we know that this can’t be the case,ymust be the true proposition and the money is behind Door B.
This is an example of a situation in which the truth of one proposition leads to the truth of another. Certainly,y can be true whenxis false; but xcan’t be true wheny is false. In this case, we say thatximpliesy.
Consider the truth table ofp→q,Table 3.1.7. Ifpimpliesq, then the third case can be ruled out, since it is the case that makes a conditional proposition false.
Definition 3.3.11 Implication. Let S be a set of propositions and let r ands be propositions generated by S. We say thatr impliess if r→ sis a tautology. We writer⇒sto indicate this implication. ♢ Example 3.3.12 Disjunctive Addition. A commonly used implication called “disjunctive addition” is p ⇒ (p∨q), which is verified by truth table
CHAPTER 3. LOGIC 49 Table 3.3.13. □ p q p∨q p→p∨q 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 1
Table 3.3.13: Truth Table to verify thatp⇒(p∨q)
If we letprepresent “The money is behind Door A” andq represent “The money is behind Door B,”p⇒(p∨q)is a formalized version of the reasoning used in Example 3.3.12. A common name for this implication is disjunctive addition. In the next section we will consider some of the most commonly used implications and equivalences.
When we defined what we mean by a Proposition Generated by a Set, we didn’t include the conditional and biconditional operators. This was because of the two equivalencesp→q⇔ ¬p∨qandp↔q⇔(p∧q)∨(¬p∧¬q). Therefore, any proposition that includes the conditional or biconditional operators can be written in an equivalent way using only conjunction, disjunction, and negation. We could even dispense with disjunction sincep∨qis equivalent to a proposition that uses only conjunction and negation.
3.3.4 A Universal Operation
We close this section with a final logical operation, the Sheffer Stroke, that has the interesting property that all other logical operations can be created from it. You can explore this operation inExercise 184.108.40.206
Definition 3.3.14 The Sheffer Stroke. The Sheffer Stroke is the logical operator defined by the following truth table:
p q p|q
0 0 1
0 1 1
1 0 1
1 1 0
Table 3.3.15: Truth Table for the Sheffer Stroke
3.3.5 Exercises for Section 3.3
1. Given the following propositions generated byp,q, andr, which are equiv- alent to one another?
(a) (p∧r)∨q (b) p∨(r∨q) (c) r∧p (d) ¬r∨p (e) (p∨q)∧(r∨q) (f) r→p (g) r∨ ¬p (h) p→r 2.
CHAPTER 3. LOGIC 50 (b) Give an example other thanxitself of a proposition generated byp,
q, andr that is equivalent tox.
(c) Give an example of a proposition other thanxthat impliesx. (d) Give an example of a proposition other thanxthat is implied byx. 3. Is an implication equivalent to its converse? Verify your answer using a
4. Suppose thatxis a proposition generated byp,q, andrthat is equivalent top∨ ¬q. Write out the truth table forx.
5. How large is the largest set of propositions generated bypandqwith the property that no two elements are equivalent?
6. Find a proposition that is equivalent top∨q and uses only conjunction and negation.
7. Explain why a contradiction implies any proposition and any proposition implies a tautology.
8. The significance of the Sheffer Stroke is that it is a “universal” operation in that all other logical operations can be built from it.
(a) Prove that p|qis equivalent to ¬(p∧q). (b) Prove that¬p⇔p|p.
(c) Build ∧using only the Sheffer Stroke. (d) Build∨using only the Sheffer Stroke.