Answer
(a) Commutative law
(b) Distributive law
(c) Negation law
(d) Identity law
Work Step by Step
We use law of logical equivalence to show (p ∨ ∼q) ∧ (∼p ∨ ∼q) ≡ ∼q.
Apply commutative law to (p ∨ ∼q) and (∼p ∨ ∼q) to get
(p ∨ ∼q) ∧ (∼p ∨ ∼q) ≡ (∼q ∨ p) ∧ (∼q ∨ ∼p).
Next apply distributive law to (∼q ∨ p) ∧ (∼q ∨ ∼p) to get
(∼q ∨ p) ∧ (∼q ∨ ∼p) ≡ ∼q ∨ (p ∧ ∼p).
Then from Negation law we know that (p ∧ ∼p) ≡ c,
so ∼q ∨ (p ∧ ∼p) ≡ ∼q ∨ c
Finally from Identity law we get ∼q ∨ c ≡ ∼q.
Hence (p ∨ ∼q) ∧ (∼p ∨ ∼q) ≡ ∼q using the 4 laws stated above.
