Using rules in logic, prove the following : p ↔ q ≡ ~ (p ∧ ~q) ∧ … - Vidyadip

Question

Using rules in logic, prove the following : p ↔ q ≡ ~ (p ∧ ~q) ∧ ~(q ∧ ~p)

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Answer

By the rules of negation of biconditional,
~(p ↔ q) ≡ (p ∧ ~q) ∨ (q ∧ ~p)
∴ ~ [(p ∧ ~ q) ∨ (q ∧ ~p)] ≡ p ↔ q
∴ ~(p ∧ ~q) ∧ ~(q ∧ ~p) ≡ p ↔ q … (Negation of disjunction)
≡ p ↔ q ≡ ~(p ∧ ~ q) ∧ ~ (q ∧ ~p).

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