Question 3508: Valid Counterexamples in a Candidate Set

Original question and givens

Using classical truth semantics, the explicit finite integer domain is U={-5, -4, -2} and P(x) means x is an integer multiple of (3). Which members of C={-7, -5, -4, -2, 2} 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={-5, -4, -2} and P(x) means x is an integer multiple of (3). Which members of C={-7, -5, -4, -2, 2} 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={-5, -4, -2};predicate=multiple 3

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

test 3 divides -5

-5=-2·3+1

x=-5;predicate=multiple 3

P(-5)=No

test 3 divides -4

-4=-2·3+2

x=-4;predicate=multiple 3

P(-4)=No

test 3 divides -2

-2=-1·3+1

x=-2;predicate=multiple 3

P(-2)=No

[[-5, No], [-4, No], [-2, No]]

all finite-domain failures={-5, -4, -2}

candidate=-7;c∈U=No

legal counterexample=No

candidate=-5;c∈U=Yes

legal counterexample=Yes

candidate=-4;c∈U=Yes

legal counterexample=Yes

candidate=-2;c∈U=Yes

legal counterexample=Yes

candidate=2;c∈U=No

legal counterexample=No

the explicitly declared finite exercise

no inference about an unspecified infinite domain

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

{-5, -4, -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.