Question 3516: Valid Counterexamples in a Candidate Set

Original question and givens

Using classical truth semantics, the explicit finite integer domain is U={3, 4, 6} and P(x) means x is an integer multiple of (2). Which members of C={1, 3, 4, 6, 10} 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={3, 4, 6} and P(x) means x is an integer multiple of (2). Which members of C={1, 3, 4, 6, 10} 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={3, 4, 6};predicate=multiple 2

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

test 2 divides 3

3=1·2+1

x=3;predicate=multiple 2

P(3)=No

test 2 divides 4

4=2·2+0

x=4;predicate=multiple 2

P(4)=Yes

test 2 divides 6

6=3·2+0

x=6;predicate=multiple 2

P(6)=Yes

[[3, No], [4, Yes], [6, Yes]]

all finite-domain failures={3}

candidate=1;c∈U=No

legal counterexample=No

candidate=3;c∈U=Yes

legal counterexample=Yes

candidate=4;c∈U=Yes

legal counterexample=No

candidate=6;c∈U=Yes

legal counterexample=No

candidate=10;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

{3}

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.