# Question 4696: Q Results and Natural-System Eligibility

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0008-04696 · q-20261008-advance-20261011-r02-arithmetic-operations-0096



## Original question and givens

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=17, b=1. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.


- G001 — All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=17, b=1. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.





## 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-arithmetic-operations](https://ocme.evemisslab.com/en/mko/mko-arithmetic-operations/)





## Step-by-step derivation

S001 · impliesBefore

exact arithmetic in Q;N={0,1,2,...} when membership is asked

After

do not truncate subtraction or rational division

Conditions: none

Compute exact results in Q before testing N-membership;do not clamp negative differences to0 or replace rational division by integer division.Division by0 remains undefined.

Step identifiers and dependenciesRule: exact_rational_arithmetic_scope · Uses: G001





S002 · calculationBefore

Q difference := sub(17, 1)

After

Q difference = 16

Conditions: none

Subtract in Q,retaining a negative result.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

Q quotient := div(17, 1)

After

Q quotient = 17

Conditions: none

Use exact division only with a nonzero denominator.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

a=17;b=1;difference=16

After

(16, Yes, 17/1, Yes)

Conditions: none

Membership in N requires a nonnegative integer.An undefined quotient has no numeric N-membership verdict,so do not fill its verdict with Yes or No.

Step identifiers and dependenciesRule: rational_result_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 — Recompute Q quotient with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute Q difference with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(16, Yes, 17/1, Yes)

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: 6805cdee122139b47398a865a146edeae299d676105c71b4ae3a49167e15c444
- candidate record SHA-256: 58370758ffd89ecd4ebd4efd619d1cc8cf866e3c596fbe0f1e5f973ed43cc4c5
- semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289
- question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0096.json)