Direct proof

Summary

A direct proof of an implication $p \to q$ assumes $p$ and then gets $q$ by a chain of valid steps. The steps use axioms, definitions and theorems that are already proved.

Remarks

Example: if $n$ is odd, then $n^2$ is odd. Write $n = 2k + 1$. Then $n^2 = 2(2k^2 + 2k) + 1$, which is odd. When a direct proof is hard, a proof by contrapositive or a proof by contradiction can help.

Explanations

L1 human

In a direct proof you go straight from the hypothesis to the conclusion. You write down what the hypothesis means, then you compute or reason until you reach the conclusion.