Chapter 2 - The Logic of Compound Statements - Exercise Set 2.2 - Page 50: 30

p $\land$ (q $\lor$ r) $\leftrightarrow$ (p $\land$ q) $\lor$ (p $\land$ r) The truth table for the tautology:

Work Step by Step

To construct the truth table, first fill in the 8 possible combinations of truth values for p, q, and r. To evaluate (q ∨ r) recall the definition of OR (a ∨ b is true when either a is true, or b is true, or both a and b are true; it is false only when both a and b are false). To evaluate (p ∧ q), (p ∧ r), and p ∧ (q ∨ r) recall the definition of AND (a ∧ b is true when, and only when, both a and b are true. If either a or b is false or if both are false, a ∧ b is false). The two statements are only logically equivalent if, and only if, they have identical truth values for each possible substitution of statements for their statement variables. The truth table shows p $\land$ (q $\lor$ r) is logically equivalent to (p $\land$ q) $\lor$ (p $\land$ r). To evaluate p $\land$ (q $\lor$ r) $\leftrightarrow$ (p $\land$ q) $\lor$ (p $\land$ r), recall the definition of biconditional (a ↔ b is true if both a and b have the same truth values and is false if a and b have opposite truth values). p $\land$ (q $\lor$ r) $\leftrightarrow$ (p $\land$ q) $\lor$ (p $\land$ r) is a tautology because all of its truth values are T.

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