Proof by cases

Summary

A proof by cases splits the situation into a finite number of cases that cover all possibilities. Then it proves the statement in each case. The basic form is this rule: from $p \lor q$, $p \to r$ and $q \to r$, conclude $r$.

Remarks

Another name is proof by exhaustion, mainly when the cases are many. Example: $n^2 + n$ is even for each integer $n$. If $n$ is even, $n^2$ and $n$ are even. If $n$ is odd, $n^2$ and $n$ are odd. In both cases, the sum is even. A split into $p$ and $\neg p$ uses the law of excluded middle.

Explanations

L1 human

When one argument does not work for all cases, split the cases. Integers are even or odd, so a proof for even integers and a proof for odd integers together cover all integers. Make sure that no case is missing.