Question 4857: Exact Rational Candidates for N

On this page

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

Return to catalogue page 25

← Question derivation index · Original question batch

R0008-04857 · q-20261008-advance-20261011-r01-natural-number-0082

Original question and givens

N includes 0. Ordered rational candidates=[6/1, -6/1, 7/13]. Give the Yes/No membership list for N.

G001 — N includes 0. Ordered rational candidates=[6/1, -6/1, 7/13]. 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.

mko-natural-number

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=[6, 1];reduced=6/1

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=[-6, 1];reduced=-6/1

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=[7, 13];reduced=7/13

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: 2b4221dbccf2d967affd8dbef98cdc981442660073d6cf6470b0675fce94f3c4
candidate record SHA-256: f84169bf01f62f58a2cb6c1fe76d3a4595cc603d6289b019dcd34ccbf8e7631c
semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289
question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON