Question

Using rules in logic, prove the following:

∼ (p ∨ q) ∨ (∼p ∧ q) ≡ ∼p

Answer 1

∼ (p ∨ q) ∨ (∼p ∧ q)
≡ (∼p ∧ ∼q) ∨ (∼p ∧ q) ..........(Negation of disjunction)
≡ ∼p ∧ (∼q ∨ q) ........(Distributive Law)
≡ ∼p ∧ T ............(Complement Law)
≡ ∼p ............(Identity Law)
∴ ∼ (p ∨ q) ∨ (∼p ∧ q) ≡ ∼p