Question 3517: Valid Counterexamples in a Candidate Set

Original question and givens

Using classical truth semantics, the explicit finite integer domain is U={4, 5, 7} and P(x) means x≥5. Which members of C={2, 4, 5, 7, 11} 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={4, 5, 7} and P(x) means x≥5. Which members of C={2, 4, 5, 7, 11} 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={4, 5, 7};predicate=ge 5

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

x=4;predicate=ge 5

P(4)=No

x=5;predicate=ge 5

P(5)=Yes

x=7;predicate=ge 5

P(7)=Yes

[[4, No], [5, Yes], [7, Yes]]

all finite-domain failures={4}

candidate=2;c∈U=No

legal counterexample=No

candidate=4;c∈U=Yes

legal counterexample=Yes

candidate=5;c∈U=Yes

legal counterexample=No

candidate=7;c∈U=Yes

legal counterexample=No

candidate=11;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

{4}

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.