Without using truth table show that: p q=(p q) ( p q) - Vidyadip

Question

Without using truth table show that:
$p \leftrightarrow q=(p \wedge q) \vee(\sim p \wedge \sim q)$

✓

Answer

L.H.S = p ↔ q

≡ (p → q) ∧ (q → p) ........(Biconditional Law)

≡ (∼ p ∨ q) ∧ (∼ q ∨ p) ........(Conditional Law)

≡ [∼ p ∧ (∼ q ∨ p)] ∨ [q ∧ (∼ q ∨ p)] ....(Distributive Law)

≡ [(∼ p ∧ ∼ q)] ∨ (∼ p ∧ p)] ∨ [(q ∧ ∼ q) ∨ (q ∧ p)] .........(Distributive Law)

≡ [(∼ p ∧ ∼ q) ∨ F] ∨ [F ∨ (q ∧ p)] ........(Complement Law)

≡ (∼ p ∧ ∼ q) ∨ (q ∧ p) .......(Identity Law)

≡ (∼ p ∧ ∼ q) ∨ (p ∧ q) ........(Commutative Law)

≡ (p ∧ q) ∨ (∼ p ∧ ∼ q) ........(Commutative Law)

≡ R.H.S.

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