# Question 3517: Valid Counterexamples in a Candidate Set

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0006-03517 · q-20261003-advance-20261004-r02-counterexample-0067



## 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](https://ocme.evemisslab.com/en/mko/mko-counterexample/)





## Step-by-step derivation

S001 · impliesBefore

U={4, 5, 7};predicate=ge 5

After

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

Conditions: none

A counterexample needs both domain membership and predicate failure;an outside integer does not refute this restricted universal statement even if its formula is false.

Step identifiers and dependenciesRule: finite_counterexample_domain · Uses: G001





S002 · impliesBefore

x=4;predicate=ge 5

After

P(4)=No

Conditions: none

Substitute this integer into the declared condition;the formula's truth value does not change its domain eligibility.

Step identifiers and dependenciesRule: explicit_predicate_value · Uses: G001, S001





S003 · impliesBefore

x=5;predicate=ge 5

After

P(5)=Yes

Conditions: none

Substitute this integer into the declared condition;the formula's truth value does not change its domain eligibility.

Step identifiers and dependenciesRule: explicit_predicate_value · Uses: G001, S002





S004 · impliesBefore

x=7;predicate=ge 5

After

P(7)=Yes

Conditions: none

Substitute this integer into the declared condition;the formula's truth value does not change its domain eligibility.

Step identifiers and dependenciesRule: explicit_predicate_value · Uses: G001, S003





S005 · impliesBefore

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

After

all finite-domain failures={4}

Conditions: none

Cover the entire declared finite domain;the universal statement on empty U is vacuously true and has no counterexample.This does not extend to larger or infinite domains.

Step identifiers and dependenciesRule: complete_finite_failure_inventory · Uses: G001, S004





S006 · impliesBefore

candidate=2;c∈U=No

After

legal counterexample=No

Conditions: none

A selected candidate must lie in both C and U and fail the predicate;an outside-domain predicate is left unevaluated for this inventory.

Step identifiers and dependenciesRule: candidate_counterexample_filter · Uses: G001, S005





S007 · impliesBefore

candidate=4;c∈U=Yes

After

legal counterexample=Yes

Conditions: none

A selected candidate must lie in both C and U and fail the predicate;an outside-domain predicate is left unevaluated for this inventory.

Step identifiers and dependenciesRule: candidate_counterexample_filter · Uses: G001, S006





S008 · impliesBefore

candidate=5;c∈U=Yes

After

legal counterexample=No

Conditions: none

A selected candidate must lie in both C and U and fail the predicate;an outside-domain predicate is left unevaluated for this inventory.

Step identifiers and dependenciesRule: candidate_counterexample_filter · Uses: G001, S007





S009 · impliesBefore

candidate=7;c∈U=Yes

After

legal counterexample=No

Conditions: none

A selected candidate must lie in both C and U and fail the predicate;an outside-domain predicate is left unevaluated for this inventory.

Step identifiers and dependenciesRule: candidate_counterexample_filter · Uses: G001, S008





S010 · impliesBefore

candidate=11;c∈U=No

After

legal counterexample=No

Conditions: none

A selected candidate must lie in both C and U and fail the predicate;an outside-domain predicate is left unevaluated for this inventory.

Step identifiers and dependenciesRule: candidate_counterexample_filter · Uses: G001, S009





S011 · impliesBefore

the explicitly declared finite exercise

After

no inference about an unspecified infinite domain

Conditions: none

Complete enumeration decides this finite universal statement,but finite success or absence of counterexamples is not a universal proof over an infinite domain.

Step identifiers and dependenciesRule: finite_logic_scope · Uses: G001, S010









## 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 16ef1fbab814f0dee25f63c07dec03fd65620dd2b3405204e888dc9e17624826
- candidate record SHA-256: 8c0a6b9efc2533326e0097f5735f52da6066072ccda860160865bb4539bcd9d9
- semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
- question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-advance-20261004-r02-counterexample-0067.json)