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: 11


Truth table for (p $\rightarrow$ (q $\rightarrow$ r)) $\leftrightarrow$ ((p $\land$ q) $\rightarrow$ r):

Work Step by Step

First fill in the eight possible combinations of truth values for p, q, and r. Then valuate the if/then statement (q $\rightarrow$ r). When the if element is T and then element is F, the statement is F. In all other cases the statement is T. Next evaluate the if/then statement (p $\rightarrow$ (q $\rightarrow$ r)). Then evaluate the statement (p $\land$ q) according to the definition of AND (p $\land$ q is true when, and only when, both p and q are true. If either p or q is false, or if both are false, p $\land$ q is false). Next evaluate the if/then statement ((p $\land$ q) $\rightarrow$ r). Lastly evaluate the biconditional statement according to 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).
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