Answer
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Work Step by Step
LOGICAL EQUIVALENCES
$$p \leftrightarrow q \equiv (p \land q) \lor (\neg p \land \neg q).....(1)
$$
$$ p \rightarrow q \equiv \neg q \lor q ....(2)$$
Identity laws:
$$ p \land T \equiv p$$$$ p \lor F \equiv p$$
Commutative laws:
$$ p\lor q\equiv q \lor p$$$$ p\land q\equiv q \land p$$
Negation Laws:
$$ p \lor \neg p \equiv T$$$$ p \land \neg p \equiv F$$
Solution:
Use Logical equivalence (1):
$$ p\leftrightarrow q \equiv (p \land q)\lor (\neg p \land \neg q)$$
Use Distributive Law:
$$\equiv [p\lor (\neg p \land \neg q)] \land [q \lor (\neg p \land \neg q)]$$
$$\equiv [(p\lor \neg p) \land (p \lor \neg q)] \land [(q \lor \neg p )\land (q \lor \neg q)]$$
Use Negation Laws:
$$\equiv [T \land (p \lor \neg q)] \land [(q \lor \neg p )\land T]$$
Use Identity Law:
$$\equiv (p \lor \neg q) \land (q \lor \neg p)$$
Use Commutative Law:
$$\equiv (q \lor \neg p) \land (p \lor \neg q) $$$$\equiv (\neg p \lor q) \land (\neg q \lor p ) $$
Use Logical equivalence (2):
$$ \equiv (p \rightarrow q) \land (q \rightarrow p)$$
We have thus derived that $p \leftrightarrow q $ is logically equivalent with $ \equiv (p \rightarrow q) \land (q \rightarrow p)$
