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Question 3501: Valid Counterexamples in a Candidate Set

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0006-03501 · q-20261003-advance-20261004-r02-counterexample-0051

Original question and givens

Using classical truth semantics, the explicit finite integer domain is U=∅ and P(x) means x=0. Which members of C={-2, 0, 4} 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=∅ and P(x) means x=0. Which members of C={-2, 0, 4} 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.

Step-by-step derivation

S001 · implies

Before

U=∅;predicate=eq 0

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

[]

After

all finite-domain failures=∅

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, S001

S003 · implies

Before

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 dependencies

Rule: candidate_counterexample_filter · Uses: G001, S002

S004 · implies

Before

candidate=0;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, S003

S005 · implies

Before

candidate=4;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, S004

S006 · 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, S005

Reverse checks

  • 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

∅

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: c60bf131f9dbddf616daa21da8f2d98cd16b81257a71e98ab801e735b0a96ad8
  • candidate record SHA-256: 9736d2b2d40c9081577f3d699ebb958a2e03be7a48e0044c6a9b106d27c2c33b
  • semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
  • question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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