Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

Show that each of these conditional statements is a tautology by using truth tables.
(a) $(p \land q) \to p$ (b) $p \to (p \lor q)$
(c) role="math" localid="1668154499853" $\neg p \to (p \to q)$ (d) $(p \land q) \to (p \to q)$
(e) $\neg (p \to q) \to p$ (f) $\neg (p \to q) \to \neg q$

It is shown that the conditional statement $\neg (p \to q) \to \neg q$ is a tautology as the output of the truth table consists of only T.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Definition of Truth Tables

A logic gate truth table depicts each feasible input sequence to the gate or circuit, as well as the resulting output based on the combination of these inputs.

To Show conditional statement is a tautology using a truth table

(a) $(p \land q) \to p$

Prepare the truth table for $(p \land q) \to p .$

p	q	localid="1668156225147" $p \land q$	$(p \land q) \to p$
T	T	T	T
T	F	F	T
F	T	F	T
F	F	F	T

Truth Table

The conditional statement $(p \land q) \to p$ is a tautology as the output of the truth table consists of only T.

(b) $p \to (p \lor q)$

Prepare the truth table for $p \to (p \lor q)$ .

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

Truth Table

The conditional statement is a tautology as the output of the truth table consists of only T.

Prepare the truth table for $\neg p \to (p \to q)$

p	q	$\neg p$	$p \to q$	$\neg p \to (p \to q)$
T	T	F	T	T
T	F	F	F	T
F	T	T	T	T
F	F	T	T	T

Truth Table

The conditional statement $\neg p \to (p \to q)$ is a tautology as the output of the truth table consists of only T.

(d) $(p \land q) \to (p \to q)$

Prepare the truth table for localid="1668162884798" $(p \land q) \to (p \to q) .$

p	q	$p \land q$	$p \to q$	localid="1668163037369" $(p \land q) \to (p \to q)$
T	T	T	T	T
T	F	F	F	T
F	T	F	T	T
F	F	F	T	T

Truth Table

The conditional statement $(p \land q) \to (p \to q)$ is a tautology as the output of the truth table consists of only T.

(e) $\neg (p \to q) \to p$

Prepare the truth table for $\neg (p \to q) \to p$ .

p	q	$p \to q$	$\neg (p \to q)$	$\neg (p \to q) \to p$
T	T	T	F	T
T	F	F	T	T
F	T	T	F	T
F	F	T	F	T

Truth Table

The conditional statement $\neg (p \to q) \to \neg q$ is a tautology as the output of the truth table consists of only T.

(f) $\neg (p \to q) \to \neg q$

Prepare the truth table for $\neg (p \to q) \to \neg q$ .

p	q	$p \to q$	$\neg (p \to q)$	$\neg (p \to q) \to \neg q$
T	T	T	F	T
T	F	F	T	T
F	T	T	F	T
F	F	T	F	T

Truth Table

The conditional statement $\neg (p \to q) \to \neg q$ is a tautology as the output of the truth table consists of only T.

Therefore, it has been shown that the given conditional statements are tautologies using truth tables.

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Short Answer

Show that each of these conditional statements is a tautology by using truth tables.
(a) $(p \land q) \to p$ (b) $p \to (p \lor q)$
(c) role="math" localid="1668154499853" $\neg p \to (p \to q)$ (d) $(p \land q) \to (p \to q)$
(e) $\neg (p \to q) \to p$ (f) $\neg (p \to q) \to \neg q$

Step by Step Solution

Definition of Truth Tables

To Show conditional statement is a tautology using a truth table

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Step by Step Solution

Definition of Truth Tables

To Show conditional statement is a tautology using a truth table

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Show that each of these conditional statements is a tautology by using truth tables.
(a) $(p \land q) \to p$ (b) $p \to (p \lor q)$
(c) role="math" localid="1668154499853" $\neg p \to (p \to q)$ (d) $(p \land q) \to (p \to q)$
(e) $\neg (p \to q) \to p$ (f) $\neg (p \to q) \to \neg q$