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Recipe

Decide an existential claim over a finite domain

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

Goal: a settled verdict: true with one witness in hand, or false only after every domain element has been ruled out

Taught in The existential quantifier

Quick recipe

Search for one witness, trying likely candidates first
The moment a candidate works, stop and report it
If every element fails, the claim is false

1
Test values from the domain, starting with the ones most likely to satisfy $P(x)$ : small numbers, obvious cases, or values the predicate's structure suggests.

Example

Domain $\{1,\ldots,6\}$ , claim $\exists x, x^2>30$ . Squares grow fastest at the largest value, so test $x=6$ first: $6^2=36$ , and $36>30$ is true.

Why: starting with a promising value often finds a witness fast, since only one success is ever needed.
2
As soon as one element makes $P(x)$ true, the existential claim is true. No other element needs checking, no matter how many would have failed.

Example

$x=6$ already gives $36>30$ , true, so the search stops immediately: $\exists x \in \{1,\ldots,6\}, x^2>30$ is true, witnessed by $x=6$ .
- this candidate satisfies $P(x)$ — stop here; the claim is true, with this element as the witness.
- this candidate fails $P(x)$ — try the next candidate and repeat this step.
Why: existence needs only one success, so checking the rest of the domain afterward would be wasted work.
3
Only once the domain is finite and every single element has been checked without success can the claim be declared false.

Example

For $\exists x \in \{1,2,3,4\}, x^2=20$ : $1^2=1$ , $2^2=4$ , $3^2=9$ , $4^2=16$ . None equal $20$ , so after exhausting the domain, the claim is false.

Why: an existential claim over a finite domain is false only when the search has covered literally every element with no success.

Common mistakes

Declaring an existential claim false after checking only some of the domain, not literally every element.
Continuing to test more candidates after already finding one witness -- unnecessary once the claim is settled.
Confusing "no witness found yet" with "no witness exists" before the whole finite domain has actually been exhausted.
Reporting the witness as the predicate itself, like $P(6)$ , instead of the value that makes it work, like $x=6$ .