Question 3561: Least Counterexample in a Finite Domain

Original question and givens

Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 1, 3} and P(x) means x=-1. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

• G001 — Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 1, 3} and P(x) means x=-1. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-counterexample

Step-by-step derivation

U={-2, -1, 1, 3};predicate=eq -1

a counterexample requires c∈U and P(c)=False

x=-2;predicate=eq -1

P(-2)=No

x=-1;predicate=eq -1

P(-1)=Yes

x=1;predicate=eq -1

P(1)=No

x=3;predicate=eq -1

P(3)=No

[[-2, No], [-1, Yes], [1, No], [3, No]]

all finite-domain failures={-2, 1, 3}

{-2, 1, 3}

-2

the explicitly declared finite exercise

no inference about an unspecified infinite domain

Reverse checks

• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

-2

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.