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: 50

Answer

See step by step answer for solution.

Work Step by Step

a) Two propositions are logically equivalent if they have the same truth value for any combination of truth values of the variables. A conjunction $p \downarrow q$ is true if and only if both p and q are false. A conjunction $\neg p $ is true if p is false. $\underline{p \quad p\downarrow p \quad \neg p}$ $T \quad F\quad\quad\quad F$ $F \quad T\quad\quad\quad T$ Since the last two columns of the truth table contain the same truth value in every row, the last two expressions are logically equivalent. b) $\underline{p \quad q \quad p\downarrow q \quad (p\downarrow q)\downarrow (p\downarrow q) \quad p \lor q}$ $T \quad T \quad F \quad\quad\quad\quad\quad T\quad\quad\quad\quad \quad T$ $T \quad F \quad F \quad\quad\quad\quad\quad T\quad\quad\quad\quad \quad T$ $F \quad T \quad F \quad\quad\quad\quad\quad T\quad\quad\quad\quad \quad T$ $F \quad F \quad T \quad\quad\quad\quad\quad F\quad\quad\quad\quad \quad F$ Since the last two columns of the truth table contain the same truth value in every row, the last two expressions are logically equivalent. c) The set {$ \neg, \lor $} if functionally complete and we have proved that both $\neg$ and $ \lor$ can be represented with $\downarrow$ ($ \neg p \equiv p \downarrow p$ and $p \lor q \equiv (p\downarrow q)\downarrow (p\downarrow q)$). So, we conclude that {$\downarrow$} is a functionally complete collection.
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