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Q30E.

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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 (p∨q)∧(¬p∨r)→(q∨r) is a tautology

(p∨q)∧(¬p∨r)→(q∨r) is a tautology.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

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.

Most popular questions for Math Textbooks

Show that (s*)*=swhen is a compound proposition

Let p, q, and r be the propositions

p : You have the flu.

q: You miss the final examination.

r : You pass the course.

Express each of these propositions as an English sentence.

a) p→q

b) ¬q↔r

c) q→¬r

d) p∨q∨r

e) (p→¬r)∨(q→¬r)

f ) (p∧q)∨(¬q∧r)

Let p and q be the propositions

p : You drive over 65 miles per hour.

q : You get a speeding ticket. Write these propositions using p and q and logical connectives (including negations).

a) You do not drive over 65 miles per hour.

b) You drive over 65 miles per hour, but you do not get a speeding ticket.

c) You will get a speeding ticket if you drive over 65 miles per hour.

d) If you do not drive over 65 miles per hour, then you will not get a speeding ticket.

e) Driving over 65 miles per hour is sufficient for getting a speeding ticket.

f) You get a speeding ticket, but you do not drive over 65 miles per hour.

g) Whenever you get a speeding ticket, you are driving over 65 miles per hour.

Show that (p→q)→r and p→(q→r)are not logically equivalent

A says “We are both knaves” and B says nothing. Exercises 24–31 relate to inhabitants of an island on which there are three kinds of people: knights who always tell the truth, knaves who always lie, and spies (called normals by Smullyan [Sm78]) who can either lie or tell the truth. You encounter three people, A, B, and C. You know one of these people is a knight, one is a knave, and one is a spy. Each of the three people knows the type of person each of other two is. For each of these situations, if possible, determine whether there is a unique solution and determine who the knave, knight, and spy are. When there is no unique solution, list all possible solutions or state that there are no solutions.

A says “I am the knave,” B says “I am the knave,” and C says “I am the knave.”
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