Question 5192: Non-Strict and Strict Inclusion Witnesses

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0009-05192 · q-20261008-advance-20261011-r05-subset-0042

Original question and givens

Finite integer sets A=[-16, 0] and B=[0]; 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=[-16, 0] and B=[0]; 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

Step-by-step derivation

S001 · implies

Before

A=[-16, 0];B=[0];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=-16;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 dependencies

Rule: inclusion_obstruction_row · Uses: G001, S001

S003 · implies

Before

A\B: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 dependencies

Rule: inclusion_obstruction_row · Uses: G001, S002

S004 · implies

Before

B\A:source element=0;target=[-16, 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 dependencies

Rule: inclusion_obstruction_row · Uses: G001, S003

S005 · implies

Before

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

After

non-strict=No;strict=No

Conditions: none

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

Step identifiers and dependencies

Rule: 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

(No, No, [-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: 6acce5472a6417a0897d14a4e6651b5884ce4d7d2011bf5ccf2fe6c8d0bf08ad
candidate record SHA-256: 8b2e699e48dabaa2dc8454f1f6df484a0370230375404d9cb3cc195ee44de695
semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56
question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462
Lean: NOT_RUN · human review: NOT_PERFORMED

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