# Question 5182: Non-Strict and Strict Inclusion Witnesses

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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.

R0009-05182 · q-20261008-advance-20261011-r05-subset-0032



## Original question and givens

Finite integer sets A=[0] and B=[0, 6]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.


- G001 — Finite integer sets A=[0] and B=[0, 6]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.





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





## Step-by-step derivation

S001 · impliesBefore

A=[0];B=[0, 6];same integer type

After

A⊆B means every A-element belongs to B

Conditions: none

An inclusion obstruction must belong to A but not B;an element outside A is no witness,and empty A gives no obstruction.

Step identifiers and dependenciesRule: set_inclusion_definition · Uses: G001





S002 · impliesBefore

A\B:source element=0;target=[0, 6]

After

obstruction=No

Conditions: none

Audit every source-set element;list it as an obstruction exactly when the target omits it.

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





S003 · impliesBefore

B\A:source element=0;target=[0]

After

obstruction=No

Conditions: none

Audit every source-set element;list it as an obstruction exactly when the target omits it.

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





S004 · impliesBefore

B\A:source element=6;target=[0]

After

obstruction=Yes

Conditions: none

Audit every source-set element;list it as an obstruction exactly when the target omits it.

Step identifiers and dependenciesRule: inclusion_obstruction_row · Uses: G001, S003





S005 · impliesBefore

A\B=[];B\A=[6]

After

non-strict=Yes;strict=Yes

Conditions: none

Strict inclusion requires A⊆B plus a B-element outside A;equal sets satisfy only non-strict inclusion.

Step identifiers and dependenciesRule: strict_subset_extra_witness · Uses: G001, S004









## Reverse checks


- 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

(Yes, Yes, [], [6])

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: dce5aceb09fd9b482dd6a617ff1606c09f100dda700fa545774ae4a5d66aecd2
- candidate record SHA-256: 937bb1308c7f5f84b38a1b96666a1da51dc3c63d6019eb39c8d588e45a93f3a4
- semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56
- question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261011-r05-subset-0032.json)