Direct Proof

Start from the hypothesis, unfold the definitions, and push the algebra until the conclusion has nowhere left to hide.

The idea

A direct proof establishes a statement of the form if $P$, then $Q$ by the most straightforward route: assume $P$, and reason forward until $Q$ appears. Such a statement claims nothing about the cases where $P$ is false, so this is all an implication requires — that whenever $P$ holds, $Q$ holds as well.

The working step is to unfold each term of $P$ into its definition, because the definition is what turns a claim into something we can compute with. "$x$ is rational" admits no algebra; "$x = p/q$ for integers $p$ and $q$ with $q \neq 0

Read $Q$ through its definition as well: the proof is finished when it produces exactly the object $Q$ asks for, not something that merely looks close. If $Q$ says a number is divisible by $d$, the proof must exhibit the integer that the number is $d$ times.

Ways to work on it

• Walkthrough. See the basic structure of a direct proof on parity and divisibility.
• Practice. Short algebra drills that mimic direct-proof moves.
• Hardest. Prove transitivity of divisibility from the definition.

Not sure where to start? Take the ten-question placement test.