Question

If q is false and p ∧ q ↔ r is true, then which one of the following statements is a tautology?
1. p ∨ r
2. (p ∧ r) → (p ∨ r)
3. (p ∨ r) → (p ∧ r)
4. p ∧ r

Answer 1

Correct Answer - Option 2 : (p ∧ r) → (p ∨ r)

From question, the statement given is:

p ∧ q ↔ r is true

This is possible under two cases.

Case 1: When both p ∧ q and r are true, which is not possible because q is false.

Case 2: When both (p ∧ q) and r are false.

p ≡ T or F; q ≡ F, r ≡ F

Now, from options:

(a) p ∨ r is T or F.

(b) (p ∧ r) → (p ∨ r) is F→(T or F), which always results in T.

(c) (p ∨ r) → (p ∧ r) is (T or F) → F, which may be T or F.

(d) p ∧ r is F.