Question 3524: Valid Counterexamples in a Candidate Set

Original question and givens

Using classical truth semantics, the explicit finite integer domain is U={11, 12, 14} and P(x) means x is an integer multiple of (4). Which members of C={9, 11, 12, 14, 18} 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={11, 12, 14} and P(x) means x is an integer multiple of (4). Which members of C={9, 11, 12, 14, 18} 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={11, 12, 14};predicate=multiple 4

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

test 4 divides 11

11=2·4+3

x=11;predicate=multiple 4

P(11)=No

test 4 divides 12

12=3·4+0

x=12;predicate=multiple 4

P(12)=Yes

test 4 divides 14

14=3·4+2

x=14;predicate=multiple 4

P(14)=No

[[11, No], [12, Yes], [14, No]]

all finite-domain failures={11, 14}

candidate=9;c∈U=No

legal counterexample=No

candidate=11;c∈U=Yes

legal counterexample=Yes

candidate=12;c∈U=Yes

legal counterexample=No

candidate=14;c∈U=Yes

legal counterexample=Yes

candidate=18;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

{11, 14}

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.