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Question 5130: Three Ordered Inclusion Obstructions

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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R0009-05130 · q-20261008-advance-20261011-r05-subset-0055

Original question and givens

Finite integer sets A=[-4, 0] and B=[0, 4]; repeated elements count once. C=[0, 5] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.

  • G001 — Finite integer sets A=[-4, 0] and B=[0, 4]; repeated elements count once. C=[0, 5] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.

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

A=[-4, 0];B=[0, 4];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 dependencies

Rule: set_inclusion_definition · Uses: G001

S002 · implies

Before

A\B:source element=-4;target=[0, 4]

After

obstruction=Yes

Conditions: none

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

Step identifiers and dependencies

Rule: inclusion_obstruction_row · Uses: G001, S001

S003 · implies

Before

A\B:source element=0;target=[0, 4]

After

obstruction=No

Conditions: none

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

Step identifiers and dependencies

Rule: inclusion_obstruction_row · Uses: G001, S002

S004 · implies

Before

B\C:source element=0;target=[0, 5]

After

obstruction=No

Conditions: none

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

Step identifiers and dependencies

Rule: inclusion_obstruction_row · Uses: G001, S003

S005 · implies

Before

B\C:source element=4;target=[0, 5]

After

obstruction=Yes

Conditions: none

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

Step identifiers and dependencies

Rule: inclusion_obstruction_row · Uses: G001, S004

S006 · implies

Before

A\C:source element=-4;target=[0, 5]

After

obstruction=Yes

Conditions: none

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

Step identifiers and dependencies

Rule: inclusion_obstruction_row · Uses: G001, S005

S007 · implies

Before

A\C:source element=0;target=[0, 5]

After

obstruction=No

Conditions: none

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

Step identifiers and dependencies

Rule: inclusion_obstruction_row · Uses: G001, S006

S008 · implies

Before

([-4], [4], [-4])

After

three independently audited directional differences

Conditions: none

If both first inclusions hold,the third follows.But a failed link does not determine the third obstruction set;use the explicit sets to audit it.

Step identifiers and dependencies

Rule: inclusion_chain_scope · Uses: G001, S007

Reverse checks

  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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

([-4], [4], [-4])

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: 4920a27cdade4696926a276e617834f0026afecc862cf11daad0ccc59ec0f80c
  • candidate record SHA-256: 1fbd9240463bb51ac2f5e9fc5f8635aff224dc86253d5826064ebaa6865796e5
  • semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56
  • question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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