Question

1.    Show that are logically equivalent. P ↔ Q and (P ∧ Q) V ( ¬P ∧ ¬Q)

Answer 1

Let us show that $p ↔q$ and $(p ∧ q) ∨ (¬p ∧ ¬q)$ are logically equivalent. It follows that

$p ↔ q=(p\to q)\land (q\to p) \\=(\neg p\lor q)\land (\neg q\lor p) \\=(\neg p\land\neg q)\lor(\neg p\land p)\lor(q\land\neg q)\lor(q\land p) \\=(\neg p\land\neg q)\lor F\lor F\lor(q\land p) \\=(\neg p\land\neg q)\lor (p\land q) \\= (p\land q)\lor (\neg p\land\neg q)$