Without using truth table show that - (p ˅ q) ˄ (∼p v ∼q) ≡ (p ∧ … - Vidyadip

Question

Without using truth table show that -

(p ˅ q) ˄ (∼p v ∼q) ≡ (p ∧ ∼q) ˄ (∼p ∧ q)

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Answer

(p ∨ q) ∧ (∼p ˅ ∼q)

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

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

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

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

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

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