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Q36E

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Discrete Mathematics and its Applications

Found in: Page 36

Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

Construct a truth table for each of these compound propositions.
a) $(p \lor q) \lor r$
b) $(p \lor q) \land r$
c) $(p \land q) \lor r$
d) $(p \land q) \lor r$
e) $(p \lor q) \land \neg r$
f) $(p \land r) \lor \neg r$

a)

p	q	r	$p \lor q$	$(p \lor q) \lor r$
T	T	T	T	T
T	T	F	T	T
T	F	T	T	T
T	F	F	T	T
F	T	T	T	T
F	T	F	T	T
F	F	T	F	T
F	F	F	F	F

b)

p	q	r	$p \lor q$	$(p \land r) \lor r$
T	T	T	T	T
T	T	F	T	F
T	F	T	T	T
T	F	F	T	F
F	T	T	T	T
F	T	F	T	F
F	F	T	F	F
F	F	F	F	F

c)

p	q	r	$p \land q$	$(p \land q) \lor r$
T	T	T	T	T
T	T	F	T	T
T	F	T	F	T
T	F	F	F	F
F	T	T	F	T
F	T	F	F	F
F	F	T	F	T
F	F	F	F	F

d)

p	q	r	$p \land q$	$(p \land q) \land r$
T	T	T	T	T
T	T	F	T	F
T	F	T	F	F
T	F	F	F	F
F	T	T	F	F
F	T	F	F	F
F	F	T	F	F
F	F	F	F	F

e)

p	q	r	$p \lor q$	$(p \lor q) \land \neg r$
T	T	T	T	F
T	T	F	T	T
T	F	T	T	F
T	F	F	T	T
F	T	T	T	F
F	T	F	T	T
F	F	T	F	F
F	F	F	F	F

f)

p	q	r	$p \land q$	$(p \land q) \lor \neg r$
T	T	T	T	T
T	T	F	T	T
T	F	T	F	F
T	F	F	F	T
F	T	T	F	F
F	T	F	F	T
F	F	T	F	F
F	F	F	F	T

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Definition of truth table

A truth table is a mathematical table which is used in logic.

Truth table for a)

The truth table for given statement is as follows,

p	q	r	$p \lor q$	$(p \lor q) \lor r$
T	T	T	T	T
T	T	F	T	T
T	F	T	T	T
T	F	F	T	T
F	T	T	T	T
F	T	F	T	T
F	F	T	F	T
F	F	F	F	F

Truth table for b)

The truth table for given statement is as follows,

p	q	r	$p \lor q$	$(p \lor q) \land r$
T	T	T	T	T
T	T	F	T	F
T	F	T	T	T
T	F	F	T	F
F	T	T	T	T
F	T	F	T	F
F	F	T	F	F
F	F	F	F	F

Truth table for c)

The truth table for given statement is as follows,

p	q	r	$p \land q$	$(p \land q) \lor r$
T	T	T	T	T
T	T	F	T	F
T	F	T	T	T
T	F	F	T	F
F	T	T	T	T
F	T	F	T	F
F	F	T	F	F
F	F	F	F	F

Truth table for d)

The truth table for given statement is as follows,

p	q	r	$p \land q$	$(p \land q) \land r$
T	T	T	T	T
T	T	F	T	F
T	F	T	F	F
T	F	F	F	F
F	T	T	F	F
F	T	F	F	F
F	F	T	F	F
F	F	F	F	F

Truth table for e)

The truth table for given statement is as follows,

p	q	r	$p \lor q$	$(p \lor q) \neg r$
T	T	T	T	F
T	T	F	T	T
T	F	T	T	F
T	F	F	T	T
F	T	T	T	F
F	T	F	T	T
F	F	T	F	F
F	F	F	F	F

Truth table for f)

The truth table for given statement is as follows,

p	q	r	$p \land q$	$(p \land q) \lor \neg r$
T	T	T	T	T
T	T	F	T	T
T	F	T	F	F
T	F	F	F	T
F	T	T	F	F
F	T	F	F	T
F	F	T	F	F
F	F	F	F	T

Most popular questions for Math Textbooks

You can see the movie only if you are over 18 years old or you have the permission of a parent. Express your answer in terms of m: “You can see the movie,” e: “You are over 18 years old,” and p: “You have the permission of a parent.”

Which of these are propositions? What are the truth values of those that are propositions?

a) Do not pass go.

b) What time is it?

c) There are no black flies in Maine.

d) $4 + x = 5$

e) The moon is made of green cheese.

f) $2^{n} ⩾ 100$

Use truth tables to verify these equivalences

A says “The two of us are both knights” and B says A “ is a knave.”

Explain, without using a truth table, why $(p \lor q \lor r) \land (\neg p \lor \neg q \lor \neg r)$ is true when at least one of p,q, and r is true and at least one is false, but is false when all three variables have the same truth value.

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