Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 2 - The Logic of Compound Statements - Exercise Set 2.2 - Page 50: 29


p $\rightarrow$ (q $\lor$ r) $\leftrightarrow$ (p $\land$ ~q) $\rightarrow$ 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 $\lor$ 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 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). To evaluate p $\rightarrow$ (q $\lor$ r) and (p $\land$ ~q) $\rightarrow$ r, recall by the definition of a conditional statement, when the if element is T and the then element is F, the statement is F. In all other cases the statement is T. 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 $\rightarrow$ (q $\lor$ r) is logically equivalent to (p $\land$ ~q) $\rightarrow$ r. To evaluate p $\rightarrow$ (q $\lor$ r) $\leftrightarrow$ (p $\land$ ~q) $\rightarrow$ r, recall the definition of biconditional (a $\leftrightarrow$ 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 $\rightarrow$ (q $\lor$ r) $\leftrightarrow$ (p $\land$ ~q) $\rightarrow$ r is a tautology because all of its truth values are T.
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