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 3^rd and 8^th 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)

4^th & 5^th columns are identical 

∴ they are logically equivalent

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

5^th column & 8^th columns are identical 

∴ p∨ (q∧r) = (p ∨ q)∧ (p ∨ r)