# Question 5033: Set Equality with Both Difference 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.

R0009-05033 · q-20261008-advance-20261011-r04-set-0033



## Original question and givens

Finite integer sets A=[-7, 0, 7] and B=[0, 7, 8]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.


- G001 — Finite integer sets A=[-7, 0, 7] and B=[0, 7, 8]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.





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





## Step-by-step derivation

S001 · impliesBefore

raw integer list=[-7, 0, 7]

After

extensional set=[-7, 0, 7]

Conditions: none

A set depends on its elements;deduplication and numeric ordering do not change membership.The original list is retained as source.

Step identifiers and dependenciesRule: set_extensionality_normalization · Uses: G001





S002 · impliesBefore

element=-7;in A=Yes;in B=No

After

in A\B=Yes;in B\A=No

Conditions: none

Check both directional differences;equal cardinalities alone do not imply equal elements.

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





S003 · impliesBefore

element=0;in A=Yes;in B=Yes

After

in A\B=No;in B\A=No

Conditions: none

Check both directional differences;equal cardinalities alone do not imply equal elements.

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





S004 · impliesBefore

element=7;in A=Yes;in B=Yes

After

in A\B=No;in B\A=No

Conditions: none

Check both directional differences;equal cardinalities alone do not imply equal elements.

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





S005 · impliesBefore

element=8;in A=No;in B=Yes

After

in A\B=No;in B\A=Yes

Conditions: none

Check both directional differences;equal cardinalities alone do not imply equal elements.

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





S006 · impliesBefore

A\B=[-7];B\A=[8]

After

sets equal=No

Conditions: none

Both differences must be empty to establish A⊆B and B⊆A,and thus extensional equality.One empty difference alone is insufficient.

Step identifiers and dependenciesRule: set_equality_two_inclusions · Uses: G001, S005









## Reverse checks


- 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

(No, [-7], [8])

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: cccbb78c1a51f80926bfaf0dbadd7e68c5f09a6f160d7b56c83f2fa9179741b7
- candidate record SHA-256: 712c1d2eb9ad500f5d1d7c6cca2d9a0ffa14a21b39780c0f61efa1acb980e585
- 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-r04-set-0033.json)