# Logic and Mathematical Statements

## If...then... statements

In general, a mathematical statement consists of two parts: the hypothesis or assumptions, and the conclusion. Most mathematical statements you will see in first year courses have the form "If A, then B" or "A implies B" or "A $\Rightarrow$ B". The conditions that make up "A" are the assumptions we make, and the conditions that make up "B" are the conclusion.

If we are going to prove that the statement "If A, then B" is true, we would need to start by making the assumptions "A" and then doing some work to conclude that "B" must also hold.

If we want to apply a statement of the form "If A, then B", then we need to make sure that the conditions "A" are met, before we jump to the conclusion "B."

For example, if you want to apply the statement "$n$ is even $\Rightarrow$ $\frac{n}{2}$ is an integer", then you need to verify that $n$ is even, before you conclude that $\frac{n}{2}$ is an integer.

In mathematics you will often encounter statements of the form "A if and only if B" or "A $\Leftrightarrow$ B". These statements are really two "if/then" statements. The statement "A if and only if B" is equivalent to the statements "If A, then B" and "If B, then A." Another way to think of this sort of statement is as an equivalence between the statements A and B: whenever A holds, B holds, and whenever B hold, A holds.

Consider the following example: "$n$ is even $\Leftrightarrow \frac{n}{2}$ is an integer". Here the statement A is "$n$ is even" and the statement B is "$\frac{n}{2}$ is an integer." If we think about what it means to be even (namely that n is a multiple of 2), we see quite easily that these two statements are equivalent: If $n=2k$ is even, then $\frac{n}{2} = \frac{2k}{2} = k$ is an integer, and if $\frac{n}{2} = k$ is an integer, then $n=2k$ so $n$ is even.