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 49: 16


These two statements are not logically equivalent. Let p represent "you paid full price" and q represent "you didn't buy it at Crown Books." Hence "If you paid full price, you didn't buy it at Crown Books" has the form p $\rightarrow$ q, and "You didn't buy it at Crown Books or you paid full price" has the form q$\lor$p. These two statements differ in their truth values (see the truth table).

Work Step by Step

To construct the truth table, first fill in the 4 combinations of truth values for p and q. Then evaluate p $\rightarrow$ q. Recall by the definition of a conditional statement, when the if element is T and then then element is F, the statement is F. In all other cases the statement is T. Lastly evaluate q$\lor$p by the definition of OR (q$\lor$ p is true when either q is true, or p is true, or both q and p are true; it is false only when both q and p are 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.
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