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Discrete Mathematics and its Applications
Found in: Page 35
Discrete Mathematics and its Applications

Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

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Short Answer

Show that (pq)(¬pr)(qr) is a tautology

(pq)(¬pr)(qr) is a tautology.

See the step by step solution

Step by Step Solution

Step1:Definition of Tautology

Tautology results in true.

Step 2: The given statement is a tautology

We write the given statement,

\(\begin{array}{l}(p \vee q) \wedge (\neg p \vee r) \to (q \vee r)\;\, = (p \vee q) \wedge (\neg (\neg p \vee r) \vee (q \vee r))\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = (p \vee q) \wedge (p \wedge \neg r) \vee (q \vee r)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = (p \vee q) \wedge \neg (p \wedge \neg r) \wedge \neg (q \vee r)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = (p \vee q) \wedge (\neg p \vee r) \wedge (\neg q \vee \neg r)\end{array}\)

\(\begin{array}{l}(p \vee q) \wedge (\neg p \vee r) \to (q \vee r)\,\; = (p \vee \neg p) \wedge (q \vee \neg q) \vee (\neg r \vee r)......\left( {Associativity and commutativity} \right)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = T \wedge T \wedge T\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = T\end{array}\)

Hence,\((p \vee q) \wedge (\neg p \vee r) \to (q \vee r)\)is a tautology.

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