GATE | GATE-CS-2017 (Set 1) | Question 18

The statement (¬ p) → (¬ q) is logically equivalent to which of the statements below?

I. p → q

II. q → p

III. (¬ q) ∨ p

IV. (¬ p) ∨ q
(A) I only
(B) I and IV only
(C) II only
(D) II and III only


Answer: (D)

Explanation: Given,

(¬ p) → (¬ q)
= ¬ (¬ p) ∨ (¬ q) { since x → y = ¬ (x) ∨ y }
= (¬ ¬ p) ∨ (¬ q)
= (p) ∨ (¬ q) {Using double negation rule}
= (¬ q) ∨ (p) (it is equivalent to statement (iii))
= (q) → (p) { since x → y = ¬ (x) ∨ y }
(it is equivalent to statement (ii))

So, option (D) is correct.

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