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Question 5041: Set Equality with Both Difference Witnesses

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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R0009-05041 · q-20261008-advance-20261011-r04-set-0041

Original question and givens

Finite integer sets A=[-15, 0, 15] and B=[0, 15, 16]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

  • G001 — Finite integer sets A=[-15, 0, 15] and B=[0, 15, 16]; 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.

Step-by-step derivation

S001 · implies

Before

raw integer list=[-15, 0, 15]

After

extensional set=[-15, 0, 15]

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 dependencies

Rule: set_extensionality_normalization · Uses: G001

S002 · implies

Before

element=-15;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 dependencies

Rule: two_sided_set_difference_row · Uses: G001, S001

S003 · implies

Before

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 dependencies

Rule: two_sided_set_difference_row · Uses: G001, S002

S004 · implies

Before

element=15;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 dependencies

Rule: two_sided_set_difference_row · Uses: G001, S003

S005 · implies

Before

element=16;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 dependencies

Rule: two_sided_set_difference_row · Uses: G001, S004

S006 · implies

Before

A\B=[-15];B\A=[16]

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 dependencies

Rule: 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, [-15], [16])

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: e8654af7c4c90638b6fc13034a7e5ad78196560e9c1277d024e73db433e3047b
  • candidate record SHA-256: 62337da89d0e6a742597113a9e67632ec58671afbc17f8ee40a6d6ce571283ed
  • semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56
  • question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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