Vacuous Proof

Definition

If we can show that $p$ is always false, then $p\to q$ is always true.

Example: Prove $P(n)$ where $P(n)$ is “If $n$ is an integer with $10\le n\le 15$, which is a perfect square then $n$ is a perfect cube.”

Proof:

$n$ is both $10\le n\le 15$ There are no perfect squares between 10 and 15 $\therefore$ the statement that $n$ is an integer with $10\le n\le 15$, which is a perfect square, then $n$ is false for all integers $n$. $\therefore P(n)$ is true for all integers $n$.

note: this one lowkey explained mad weirdly, i’d recommend just watching a youtube video on it: Vacuous Proofs