Question 3521: Valid Counterexamples in a Candidate Set

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0006-03521 · q-20261003-advance-20261004-r02-counterexample-0071

Original question and givens

Using classical truth semantics, the explicit finite integer domain is U={8, 9, 11} and P(x) means x≥9. Which members of C={6, 8, 9, 11, 15} 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={8, 9, 11} and P(x) means x≥9. Which members of C={6, 8, 9, 11, 15} 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

S001 · implies

Before

U={8, 9, 11};predicate=ge 9

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 dependencies

Rule: finite_counterexample_domain · Uses: G001

S002 · implies

Before

x=8;predicate=ge 9

After

P(8)=No

Conditions: none

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

Step identifiers and dependencies

Rule: explicit_predicate_value · Uses: G001, S001

S003 · implies

Before

x=9;predicate=ge 9

After

P(9)=Yes

Conditions: none

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

Step identifiers and dependencies

Rule: explicit_predicate_value · Uses: G001, S002

S004 · implies

Before

x=11;predicate=ge 9

After

P(11)=Yes

Conditions: none

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

Step identifiers and dependencies

Rule: explicit_predicate_value · Uses: G001, S003

S005 · implies

Before

[[8, No], [9, Yes], [11, Yes]]

After

all finite-domain failures={8}

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 dependencies

Rule: complete_finite_failure_inventory · Uses: G001, S004

S006 · implies

Before

candidate=6;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 dependencies

Rule: candidate_counterexample_filter · Uses: G001, S005

S007 · implies

Before

candidate=8;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 dependencies

Rule: candidate_counterexample_filter · Uses: G001, S006

S008 · implies

Before

candidate=9;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 dependencies

Rule: candidate_counterexample_filter · Uses: G001, S007

S009 · implies

Before

candidate=11;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 dependencies

Rule: candidate_counterexample_filter · Uses: G001, S008

S010 · implies

Before

candidate=15;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 dependencies

Rule: candidate_counterexample_filter · Uses: G001, S009

S011 · implies

Before

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 dependencies

Rule: 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

{8}

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: 4fa3fefc6e564a0d4d3b4dfcfea17ce4e58a2417aaaea120a85325dd00754973
candidate record SHA-256: 3202a1223c627859f7bec216390f1f74a3b53dca36f41eb89aba49d2ee366362
semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
Lean: NOT_RUN · human review: NOT_PERFORMED

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