Construct a truth table to show if (p → q) ^ (p → r) and p → (q A r) are logiclly equivalent.p →r (p → q) ^ (p → r) q Arp → (q ^r)Are (p → q) A (p → r) and p → (q ^ r) logiclly equivalent?YesNoNot sure

Truth table for implication: A…

Answered: Construct a truth table to show if (p →…

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Construct a truth table to show if (p → q) ^ (p → r) and p → (q A r) are logiclly equivalent.
p →r (p → q) ^ (p → r) q Ar
p → (q ^r)
Are (p → q) A (p → r) and p → (q ^ r) logiclly equivalent?
Yes
No
Not sure

Expert Solution

Step 1

Truth table for implication:

A	B	A $\to$ B
T	T	T
T	F	F
F	T	T
F	F	T

Truth table for conjunction:

A	B	A $\land$ B
T	T	T
T	F	F
F	T	F
F	F	F

The required truth table:

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

In the above truth table we can clearly see that 6th column (i.e. (p $\to$ q) $\land$ (p $\to$ r) ) and 8th column (i.e. p $\to$ (q $\land$ r) ) are identical.

Therefore, they are logically equivalent.

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