Simple Proof Methods

Here is another type of a relatively easy proof method.

A vacuous proof is a proof in which the antecedent is always false. Since an implication "if $p$, then $q$" is always true when $p$ is false, the proof holds without needing to verify the truthfulness of $q$.

Example: Prove that for any integer $n$, if $n$ is both even and odd, then $n^3 - 5n = 42$.

Proof. The antecedent "$n$ is both even and odd" is false because no integer can be both even and odd. Since an implication with a false antecedent is always true, the statement is vacuously true.◼

Fun class activities:

1. Prove that if $x = 1$ is an even integer, then $x = 0$.
2. Prove that if $n \in \mathbb{N}$, then $2 + 10 = 12$.