Discrete Mathematics and Its Applications, Seventh Edition

Published by McGraw-Hill Education
ISBN 10: 0073383090
ISBN 13: 978-0-07338-309-5

Chapter 1 - Section 1.3 - Propositional Equivalences - Exercises - Page 36: 54

Answer

$p|(q|r)$ and $(p|q)|r$ are not logically equivalent.

Work Step by Step

Two propositions are logically equivalent if they have the same truth value for any combination of truth values of the variables. p NAND q is true if and only if both p or q or both are false. p NAND q is denoted as p|q $\underline{p\quad q \quad r\quad q|r\quad p|q\quad p|(q|r)\quad (p|q)|r}$ $T\quad T \quad T\quad F\quad F\quad\quad T\quad\quad\quad T$ $T\quad T \quad F\quad T\quad F\quad\quad F\quad\quad\quad T$ $T\quad F \quad T\quad T\quad T\quad\quad F\quad\quad\quad F$ $T\quad F \quad F\quad T\quad T\quad\quad F\quad\quad\quad T$ $F\quad T \quad T\quad F\quad T\quad\quad T\quad\quad\quad F$ $F\quad T \quad F\quad T\quad T\quad\quad T\quad\quad\quad T$ $F\quad F \quad T\quad T\quad T\quad\quad T\quad\quad\quad F$ $F\quad F \quad F\quad T\quad T\quad\quad T\quad\quad\quad T$ Since the last two columns of the truth table $\underline{do not}$ contain the same truth value in every row, the last two expressions are NOT logically equivalent. Hence the logical operator | is not associative.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.