Showing $P \leftrightarrow Q \iff (P \lor Q) \to (P \land Q)$ using propositional logic laws

Question

$$ (P\leftrightarrow Q) \equiv (P\lor Q)\to(P\land Q)$$

How can I prove this equation using propositional logic laws? it would be great if someone can explain this with all steps.

Answer 1

$P\leftrightarrow Q$ is true whenever $P$ and $Q$ have the same truth value, i.e. when they're both false or both true. Hence: \begin{align} P\leftrightarrow Q&=(\neg P\land\neg Q)\lor (P\land Q)\\ &=\neg(P\lor Q)\lor (P\land Q)\\ &=(P\lor Q)\to(P\land Q) \end{align}

First line: this is the definition of the biconditional ($\leftrightarrow$) as a disjunction between the two cases "both $P$ and $Q$ are false" or "both $P$ and $Q$ are true".

Second line: we apply De Morgan's law to the first parenthesis, "factoring out" a negation ($\neg$). Indeed, having both $P$ and $Q$ false is the same has not having either of them true.

Third line: we move from the second to the third line by applying the definition of the implication ($\to$) as a disjunction. For a premise to logically imply a conclusion, we require the conclusion to never be false when the premise is false, or equivalenty, as in our case, that either the premise be false or the conclusion be true. Here the premise is $(P\lor Q)$ and the conclusion is $(P\land Q)$.