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Question 4872: Exact Rational Candidates for N

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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R0008-04872 · q-20261008-advance-20261011-r01-natural-number-0097

Original question and givens

N includes 0. Ordered rational candidates=[21/1, -21/1, 22/43]. Give the Yes/No membership list for N.

  • G001 — N includes 0. Ordered rational candidates=[21/1, -21/1, 22/43]. Give the Yes/No membership list for N.

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

N={0,1,2,...}

After

0 belongs;negative and nonintegral rational values do not

Conditions: none

This convention includes0 in N;distinguish membership,rational values,partial predecessors and closed intervals.

Step identifiers and dependencies

Rule: natural_zero_convention · Uses: G001

S002 · implies

Before

raw rational=[21, 1];reduced=21/1

After

nonnegative integer=Yes

Conditions: none

First reduce to its exact value,then require nonnegativity and reduced denominator1.A representation whose original denominator is not1 can still be natural.

Step identifiers and dependencies

Rule: reduced_rational_natural_membership · Uses: G001, S001

S003 · implies

Before

raw rational=[-21, 1];reduced=-21/1

After

nonnegative integer=No

Conditions: none

First reduce to its exact value,then require nonnegativity and reduced denominator1.A representation whose original denominator is not1 can still be natural.

Step identifiers and dependencies

Rule: reduced_rational_natural_membership · Uses: G001, S002

S004 · implies

Before

raw rational=[22, 43];reduced=22/43

After

nonnegative integer=No

Conditions: none

First reduce to its exact value,then require nonnegativity and reduced denominator1.A representation whose original denominator is not1 can still be natural.

Step identifiers and dependencies

Rule: reduced_rational_natural_membership · Uses: G001, S003

Reverse checks

  • 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

[Yes, No, No]

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: 91fc809a757c46f2e10a8c374f4efa4db325672ea614a953b80d0f43b5bdadfc
  • candidate record SHA-256: d843983c8aa8a7413adc8d9d91428f2ff59cb85dd726b421e8bb6718ea3b7de1
  • semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289
  • question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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