Question

Examine whether the following are logically equivalent:

(i) p ↔ q and (p → q) ∧ (q → p)

(ii) p → (q → r) and (p → q) → r

(iii) (p ∧ ~q) ∨ q and p ∨ q.

(iv) p ↔ q and (~ p ∨ q) ∧ (~q ∨ p)

(v) p ∧ q and ~(p →~q)

(vi) ~ (p ↔ q ) and (p ∧ ~q) ∨ (q ∧~p)

(vii) p∨ (q ∧ r) and (p ∨ q) ∧ (p ∨ r)

Answer 1

From 3rd & 6th column we conclude that

p ↔ q (p ↔ q) ∧ (q ↔ p)

(ii) p → (q → r) and (p → q) → r

From last two columns we conclude that p → (q → r) and (p → q) → r are not logically equivalent

(iii) (p ∧ ~q) ∨ q and p ∨ q

From last two columns we conclude (p ∧ ~q) ∨ q p ∨ q.

(iv) p↔ q and (~ p ∨ q) ∧ ( ~q ∨ p)

From 3rd and 8th columns we conclude that p ↔ q ≡ (~ p ∨ q) ∧ (~q ∨ p)

(v) P ∧ q and (p → ~q)

Column 3 and column 6 are identical :

∴ They are logically equivalent

(vi) ~ (p ↔ q ) and (p ∧ ~q) ∨ (q ∧~p)

4th & 5th columns are identical

∴ they are logically equivalent

(vii) p ∨ (q ∧ r) and (p ∨ q)∧ (p ∨ r)

5th column & 8th columns are identical

∴ p∨ (q∧r) = (p ∨ q)∧ (p ∨ r)