Trivial Proof

Definition

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

Example: Let $P(n)$ be “if $a$ and $b$ are non-negative integers and $a\ge b$, then $a^n\ge b^n$. Show that $P(0)$ is true.

• What is the domain? Non-negative integers (includes zero)
• We need to show $P(0)$ is true, in other words, “if $a$ and $b$ are non-negative integers and $a\ge b$, then $a^0\ge b^0$.”

Let’s do an equivalence style proof:

$$\begin{array}{r}{a}^0\ge b^0\\ \equiv <algebra>\\ 1\ge 1\\ \equiv <algebra>\\ T\end{array}$$

The conclusion is always true regardless of what the values of a and b are. Therefore, P(0) is always true.