Question

If p and q are true and rands are false statements, then which of the following is true?

Options

• (q ∧ r) ∨ (~ p ∧ s)

• (~ p → q) ↔ (r ∧ s)

• (p → q) ∨ (r ↔ s)

• (p ∧ ~ r) ∧ (~ q ∨ s)

Answer 1

(p → q) ∨ (r ↔ s)

Explanation:

We have statements p, q → T and r, s → F

1st option,

(q ∧ r) ∨ (~ p ∧ s) ≡ (T ∧ F) ∨ (F ∧ F)

≡ F ∨ F ≡ F

2nd option,

(~ p → q) ↔ (r ∧ s)

≡ (p ∨ q) → (r ∨ s)   .....(∵ p → q ≡ ~ p ∨ q)

≡ (~ p ∨ q) ∨ (r ∧ s) ∧ ((p ∨ q) ∨ (~ r ∧ s))   ....(∵ p ↔ q ≡ (~ p ∨ q) ∧ (p ∨ q))

≡ (~ (T ∨ T) ∨ (F ∧ F) ∧ ((T ∨ F) ∨ (~ (F ∧ F))))

≡ (F ∨ F) ∧ (T ∨ T)

≡ F ∧ T ≡ F

3rd option,

(p → q) ∨ (r ↔ s)

≡ (~ p ∨ q) ∨ ((~ r ∨ s) ∧ (r ∨ ~ s))   ....(∵ p → q ≡ ~ p ∨ q)

≡ (F ∨ T) ∨ ((T ∨ F) ∧ (F ∨ T))

≡ T ∨ (T ∧ T) ≡ T ∨ T ≡ T

4th option do similar as 1st option.