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: 15

Answer

The statements p $\rightarrow$ (q $\rightarrow$ r) and (p $\rightarrow$ q) $\rightarrow$ r are not logically equivalent. See the truth table. The two statements differ in their truth values.
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Work Step by Step

To construct the truth table, first fill in the truth values for the eight combinations of p, q, and r. Then evaluate the if then statements (q $\rightarrow$ r) and (p $\rightarrow$ q). Recall by the definition of a conditional statement, when the if element is T and then element is F, the statement is F. In all other cases the statement is T. Lastly evaluate the truth values of the full statements p $\rightarrow$ (q $\rightarrow$ r) and (p $\rightarrow$ q) $\rightarrow$ r. 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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