text: # Propositional Calculus and Boolean Algebra
  
  Learn to make compound propositions and get introduced to propositional calculus and boolean algebra.
  
  We'll cover the following
  
  ## Arithmetic expressions and equations
  
  Since we’re familiar with real numbers, we will start with the following arithmetic equality:
  
  $$$3 \times (4 + 5) = 3 \times 4 + 3 \times 5.$$$
  
  The expression on the left-hand side has three numbers: $3, 4$, and $5$. Furthermore, we have used two arithmetic operations: addition($+$) and multiplication($\times$). These operations allow us to combine two numbers to get another number as a result. The famous BODMASBODMAS stands for brackets, order, division, multiplication, addition, and subtraction. It specifies the precedence of the arithmetic operations. rule tells us that to evaluate the left-hand side, we must first add $4$ with $5$ and then multiply $3$ by the result. The final calculation yields another number, $27$, as the answer.
  
  The calculation on the right-hand side is also dictated by rules of arithmetic (including BODMAS). It also yields the same result, therefore justifying the equality.
  
  Equation (1) is just an example of the distributive law for real numbers. In general, it states that:
  
  $$$a\times (b + c) = a \times b + a \times c.$$$
  
  Equation (2) is true for any real numbers $a, b,$ and $c$.
  
  The distributive law (and other laws) allows us to manipulate algebraic equations (over real numbers).
  
  ## Logical expressions and equivalences
  
  Just as arithmetic operations like addition, subtraction, and multiplication allow us to combine numbers to create other numbers, logical operations allow us to combine propositions to create new propositions called compound propositions. Let’s start with an illustrative example. Consider the following statement:
  
  $E_1$: If Eddie gets a bonus in January, she will buy a pair of sunglasses and a new cellphone.
  
  In the above statement $E_1$, we can readily identify three atomic statements. Let’s make a list of them:
  
  1. $J:$ Eddie gets a bonus in January.
  2. $S:$ Eddie buys a pair of sunglasses (in January).
  3. $C:$ Eddie buys a new cellphone (in January).
  
  The truth value of $E_1$ depends on the truth values of $J, S$, and $C$.
  
  Get hands-on with 1200+ tech skills courses.
explanation: The text extract is an educational resource that introduces concepts in propositional calculus and Boolean algebra. It covers arithmetic expressions and equations, including the distributive law, and then transitions to logical expressions and equivalences. The extract provides an example of a logical statement and breaks it down into atomic statements to illustrate how logical operations combine propositions.
  
  The text does not contain a specific question or a question-and-answer pair.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Propositional Calculus and Boolean Algebra
  
  Learn to make compound propositions and get introduced to propositional calculus and boolean algebra.
  
  We'll cover the following
  
  ## Arithmetic expressions and equations
  
  Since we’re familiar with real numbers, we will start with the following arithmetic equality:
  
  $$$3 \times (4 + 5) = 3 \times 4 + 3 \times 5.$$$
  
  The expression on the left-hand side has three numbers: $3, 4$, and $5$. Furthermore, we have used two arithmetic operations: addition($+$) and multiplication($\times$). These operations allow us to combine two numbers to get another number as a result. The famous BODMASBODMAS stands for brackets, order, division, multiplication, addition, and subtraction. It specifies the precedence of the arithmetic operations. rule tells us that to evaluate the left-hand side, we must first add $4$ with $5$ and then multiply $3$ by the result. The final calculation yields another number, $27$, as the answer.
  
  The calculation on the right-hand side is also dictated by rules of arithmetic (including BODMAS). It also yields the same result, therefore justifying the equality.
  
  Equation (1) is just an example of the distributive law for real numbers. In general, it states that:
  
  $$$a\times (b + c) = a \times b + a \times c.$$$
  
  Equation (2) is true for any real numbers $a, b,$ and $c$.
  
  The distributive law (and other laws) allows us to manipulate algebraic equations (over real numbers).
  
  ## Logical expressions and equivalences
  
  Just as arithmetic operations like addition, subtraction, and multiplication allow us to combine numbers to create other numbers, logical operations allow us to combine propositions to create new propositions called compound propositions. Let’s start with an illustrative example. Consider the following statement:
  
  $E_1$: If Eddie gets a bonus in January, she will buy a pair of sunglasses and a new cellphone.
  
  In the above statement $E_1$, we can readily identify three atomic statements. Let’s make a list of them:
  
  1. $J:$ Eddie gets a bonus in January.
  2. $S:$ Eddie buys a pair of sunglasses (in January).
  3. $C:$ Eddie buys a new cellphone (in January).
  
  The truth value of $E_1$ depends on the truth values of $J, S$, and $C$.
  
  Get hands-on with 1200+ tech skills courses.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract is an educational resource that introduces concepts in propositional calculus and Boolean algebra. It covers arithmetic expressions and equations, including the distributive law, and then transitions to logical expressions and equivalences. The extract provides an example of a logical statement and breaks it down into atomic statements to illustrate how logical operations combine propositions.
  
  The text does not contain a specific question or a question-and-answer pair.
  
  Label: no-question