Question 3503: Valid Counterexamples in a Candidate Set

Original question and givens

Using classical truth semantics, the explicit finite integer domain is U={-10, -9, -7} and P(x) means x=-9. Which members of C={-12, -10, -9, -7, -3} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

• G001 — Using classical truth semantics, the explicit finite integer domain is U={-10, -9, -7} and P(x) means x=-9. Which members of C={-12, -10, -9, -7, -3} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

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={-10, -9, -7};predicate=eq -9

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

x=-10;predicate=eq -9

P(-10)=No

x=-9;predicate=eq -9

P(-9)=Yes

x=-7;predicate=eq -9

P(-7)=No

[[-10, No], [-9, Yes], [-7, No]]

all finite-domain failures={-10, -7}

candidate=-12;c∈U=No

legal counterexample=No

candidate=-10;c∈U=Yes

legal counterexample=Yes

candidate=-9;c∈U=Yes

legal counterexample=No

candidate=-7;c∈U=Yes

legal counterexample=Yes

candidate=-3;c∈U=No

legal counterexample=No

the explicitly declared finite exercise

no inference about an unspecified infinite domain

Reverse checks

• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

{-10, -7}

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.