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Recipe

Check a universal claim over a finite domain

Use when: you're given $\forall x\, P(x)$ over a finite, listable domain and must decide whether it's true

Goal: a settled verdict: true only if every domain element checks out, false the moment one counterexample turns up

Taught in The universal quantifier

Quick recipe

List the domain and check the most suspicious elements first
Substitute and evaluate every remaining element
Decide the verdict from what you found

1
Write out every domain element. Before working through it in order, it can help to test the largest, smallest, or otherwise unusual values first, since a false claim often fails at an edge case.

Example

Domain $\{1,2,3,4,5\}$ , claim $\forall x, x^2 < 30$ . The largest value grows fastest, so check $x=5$ first: $5^2=25$ , and $25<30$ is true.

Why: catching a fast failure early can settle the claim without scanning the whole domain, though passing at one value never proves the rest.
2
Work through the rest of the domain, substituting each value into $P(x)$ and simplifying until its truth value is clear.

Example

$x=1$ : $1<30$ , true. $x=2$ : $4<30$ , true. $x=3$ : $9<30$ , true. $x=4$ : $16<30$ , true.

Why: a universal claim requires literally every element to pass, so any one left unchecked leaves the claim unverified.
3
If every element passed, the universal claim is true. If even one element failed, that element is a counterexample, and the claim is false.

Example

All five elements of $\{1,2,3,4,5\}$ passed $x^2<30$ , so $\forall x \in \{1,\ldots,5\}, x^2<30$ is true.
- every element passed — the claim is true over this domain.
- at least one element failed — the claim is false; report that element as the counterexample.
Why: $\forall x\, P(x)$ is true exactly when its truth set equals the whole domain, and false the instant one element breaks it.

Common mistakes

Stopping after checking only a few elements and declaring the claim true without finishing the domain.
Treating one passing element as proof, the way a witness would for an existential claim, when a universal claim needs all of them.
Continuing to check further elements after already finding one that fails -- unnecessary once a counterexample turns up.
Reporting the counterexample as $P(c)$ instead of the actual failing element $c$ itself.