Proofs:Biconditionals

A biconditional of two propositions P and Q takes the form "$P$ if and only if $Q$". This can also be written as "$P\iff Q$, which is equivalent to "$P\implies Q$ and $Q\implies P$". When proving a biconditional statement, we need to prove that $P\implies Q$ and $Q\implies P$ are true. Remember that the contrapositive of a conditional is logically equivalent to the conditional. Thus, "$P\implies Q$ and $Q\implies P$" is logically equivalent to "$P\implies Q$ and $\neg P\implies \neg Q$", "$Q\implies P$ and $\neg Q\implies \neg P$", or "$\neg P\implies \neg Q$ and $\neg Q\implies \neg P$". Thus, we do have some options as to how to prove the two directions of a biconditional statement.