Simple Proof Methods

Let's prove the truthfulness of a biconditional, which will use a mix of proof methods:

Theorem. [Exercise 4 on slide 8] Consider an $n \in \mathbb{Z}$. It holds that $n = 2$ if and only if $1 < n < 3$.

Proof. Because of the words "if and only if" (you may sometimes see "iff" instead of "if and only if",) we are working with a biconditional of the form $p \leftrightarrow q$. To show it's true, we need to show both that $p \to q$ and that $q \to p$:

• [Showing that if $n = 2$, then $1 < n < 3$ using a direct proof] Assume that $n = 2$. This means that $n = 2 > 1$ and that $n = 2 < 3$. In other words, $1 < n < 3$.
• [Showing that if $1 < n < 3$, then $n = 2$ using a proof by contrapositive] Suppose that $n \in \mathbb{Z}$ and $n \neq 2$. It means that $n$ is any integer besides $2$, which, in other words, is either smaller than or equal to $1$ or larger than or equal to $3$. This means $n \le 1$ or $n \ge 3$.

because we showed that both implication directions are true, it means that the theorem is true.◼