Simple Proof Methods

All the proofs we've seen so far and wrote so far (as part of the previous class activity) belong to the category of direct proofs. In this type of proofs, we are usually told to show that some statement of the form $p \to q$ (or $q \text{ given } p$, $p \text{ implies } q$, etc.,) is true by first assuming that $p$ is true, and then, from there, showing how $q$ is true.

For example, to prove the statement

If $n$ is even and $k \in \mathbb{Z}$, then $kn$ is also even.

which is of the form $p \to q$, we first assume that $n$ is even and that $k$ is an integer (= the $p$ part, also called the antecedent: "preceding in time or order"), and, from these premises, show that $kn$ is even (= the $q$ part, also called the consequent: "following in time or order").

Although direct proofs seem to be the most intuitive and direct approach, they are not, however, applicable for all situations. Specifically, for proving certain statements, direct proofs turns out to be quite lengthy, and sometime, impossible to write. As such, we resort to a few other proof methods, depending on the given statement to prove.