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Question 3568: Least Counterexample in a Finite Domain

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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R0006-03568 · q-20261003-advance-20261004-r02-counterexample-0043

Original question and givens

Using classical truth semantics, the explicit finite integer domain is U={5, 6, 8, 10} and P(x) means x≤8. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

  • G001 — Using classical truth semantics, the explicit finite integer domain is U={5, 6, 8, 10} and P(x) means x≤8. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

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={5, 6, 8, 10};predicate=le 8

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=5;predicate=le 8

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 dependencies

Rule: explicit_predicate_value · Uses: G001, S001

S003 · implies

Before

x=6;predicate=le 8

After

P(6)=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=8;predicate=le 8

After

P(8)=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

x=10;predicate=le 8

After

P(10)=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, S004

S006 · implies

Before

[[5, Yes], [6, Yes], [8, Yes], [10, No]]

After

all finite-domain failures={10}

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, S005

S007 · implies

Before

{10}

After

10

Conditions: none

Select the numerically least legal counterexample,not the first input row;explicitly report nonexistence if there is none.

Step identifiers and dependencies

Rule: least_numeric_failure · Uses: G001, S006

S008 · 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, S007

Reverse checks

  • 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

10

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: 407245ac1e3a72ef78f346efc0a2fe4e5ccc43eea9ecb161d7cd00ff992801de
  • candidate record SHA-256: 2ceed12fed51ecfe6d41dfe0818dc54724773ddfd51714613555ec8a2c4f73b7
  • semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
  • question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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