Q52 — Engineering Mathematics

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Engineering Mathematics

Q52. Marks: +2.0 UGC NET Paper 2: Computer Science 26th Nov 2021

Which of the following is logically equivalent?

A. ¬p → (q → r) and q → (p ∨ r)

B. (p → q) → r and p → (q → r)

C. (p → q) → (r → s) and (p → r) → (q → s)

Choose the correct answer from the options given below :

1.A only ✓ Correct

2.A and B only

3.B and C only

4.A and C only

Solution

A. ¬p → (q → r) and q → (p ∨ r)
For the above, we can make a truth table and compare them:

p	q	r	¬p	q → r	¬p → (q → r)	p v r	q → (p v r)
FALSE	FALSE	FALSE	TRUE	TRUE	TRUE	FALSE	TRUE
FALSE	FALSE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
FALSE	TRUE	FALSE	TRUE	FALSE	FALSE	FALSE	FALSE
FALSE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
TRUE	FALSE	FALSE	FALSE	TRUE	TRUE	TRUE	TRUE
TRUE	FALSE	TRUE	FALSE	TRUE	TRUE	TRUE	TRUE
TRUE	TRUE	FALSE	FALSE	FALSE	TRUE	TRUE	TRUE
TRUE	TRUE	TRUE	FALSE	TRUE	TRUE	TRUE	TRUE

We can see that both are equal, hence, they are equivalent.

B. (p → q) → r and p → (q → r)
For the above, we can make a truth table and compare them:

p	q	r	p → q	(p → q) → r	q → r	p → (q → r)
FALSE	FALSE	FALSE	TRUE	FALSE	TRUE	TRUE
FALSE	FALSE	TRUE	TRUE	TRUE	TRUE	TRUE
FALSE	TRUE	FALSE	TRUE	FALSE	FALSE	TRUE
FALSE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
TRUE	FALSE	FALSE	FALSE	TRUE	TRUE	TRUE
TRUE	FALSE	TRUE	FALSE	TRUE	TRUE	TRUE
TRUE	TRUE	FALSE	TRUE	FALSE	FALSE	FALSE
TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE

We can see that both are not equal, hence, they are not equivalent.

C. (p → q) → (r → s) and (p → r) → (q → s)
For the above, we can make a truth table and compare them:

p	q	r	s	p → q	r → s	(p → q)→(r → s)	p → r	q → s	(p → r)→(q → s)
TRUE	FALSE	FALSE	FALSE	FALSE	TRUE	TRUE	FALSE	TRUE	TRUE
TRUE	FALSE	FALSE	TRUE	FALSE	TRUE	TRUE	FALSE	TRUE	TRUE
TRUE	FALSE	TRUE	FALSE	FALSE	FALSE	TRUE	TRUE	TRUE	TRUE
TRUE	FALSE	TRUE	TRUE	FALSE	TRUE	TRUE	TRUE	TRUE	TRUE
TRUE	TRUE	FALSE	FALSE	TRUE	TRUE	TRUE	FALSE	FALSE	TRUE
TRUE	TRUE	FALSE	TRUE	TRUE	TRUE	TRUE	FALSE	TRUE	TRUE
TRUE	TRUE	TRUE	FALSE	TRUE	FALSE	FALSE	TRUE	FALSE	FALSE
TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
FALSE	FALSE	FALSE	FALSE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
FALSE	FALSE	FALSE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
FALSE	FALSE	TRUE	FALSE	TRUE	FALSE	FALSE	TRUE	TRUE	TRUE
FALSE	FALSE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
FALSE	TRUE	FALSE	FALSE	TRUE	TRUE	TRUE	TRUE	FALSE	FALSE
FALSE	TRUE	FALSE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
FALSE	TRUE	TRUE	FALSE	TRUE	FALSE	FALSE	TRUE	FALSE	FALSE
FALSE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE

We can see that both are not equal, hence, they are not equivalent.

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