Does a statement need to be a biconditional statement to prove by the contrapositive I am trying to write a proof and was wondering if a then b, the converse if b then a might not be true. This leads me to wonder if the statement needs to be an if and only if statement if it can be proven by the contrapositive if not b then not a.
That is do I need a statement like this a if and only if b to prove that if not b then then not a, or is if a then b sufficient to do that.
 A: By definition, "$a \implies b$" means "not ($a$ and not $b$)". So then "(not $b$)$\implies$(not $a$)" means "not ((not $b$) and not (not $a$))". This simplifies to "not ($a$ and not $b$)", which is the same thing as "$a \implies b$". So yeah, you only need the one direction of implication.
You should probably think about this definition of $a \implies b$ if you've never seen it. Like, draw a truth table or something.
A: In general, $p\to q$ does not imply the converse, $q\to p$. For example, using a truth table, you can show that $p\to q$ is logically equivalent to the contrapositive:
$$p\to q \equiv \neg q \to \neg p$$
In other words, you do not need $q\to p$.
If you have that $p \leftrightarrow q$, then by definition you have that
$$q\to p.$$

If you are tasked with showing that $P$ implies $Q$, then one strategy is to show instead that $\neg Q$ implies $\neg P$, since $P\to Q \equiv \neg Q\to \neg P$. Thus, assume $\neg Q$ and arrive at $\neg P$. Then you have shown $P$ implies $Q$ since they are logically equivalent.
