# Question 5626: Typed Membership Queries and Truth Values

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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.

R0010-05626 · q-20261008-advance-20261012-r04-set-membership-0001



## Original question and givens

The finite type is Fin(0)=∅. Complete characteristic table bits=[] uses 0/1 membership from index 0; ordered queries=[-1, 0, 1]. Give (type-eligibility mask, membership truth values or ill-typed).


- G001 — The finite type is Fin(0)=∅. Complete characteristic table bits=[] uses 0/1 membership from index 0; ordered queries=[-1, 0, 1]. Give (type-eligibility mask, membership truth values or ill-typed).





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





## Step-by-step derivation

S001 · impliesBefore

declared finite type or bounded integer universe

After

membership truth is distinct from eligibility to ask the question

Conditions: none

Being outside the type is not the same as false membership.Check only the declared finite type or integer set,not all set theory or unrestricted comprehension.

Step identifiers and dependenciesRule: finite_membership_scope · Uses: G001





S002 · impliesBefore

query=-1;legal in Fin(0)=No

After

type mismatch

Conditions: none

Check typing before reading a bit;do not replace ill-typed with False.For Fin(0),every integer query is outside the type.

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





S003 · impliesBefore

query=0;legal in Fin(0)=No

After

type mismatch

Conditions: none

Check typing before reading a bit;do not replace ill-typed with False.For Fin(0),every integer query is outside the type.

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





S004 · impliesBefore

query=1;legal in Fin(0)=No

After

type mismatch

Conditions: none

Check typing before reading a bit;do not replace ill-typed with False.For Fin(0),every integer query is outside the type.

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









## Reverse checks


- 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, No], [ill-typed, ill-typed, ill-typed])

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: d1ffaaed51b7c91c12a431b4180c6b2da6e90320ea626bd59acca18d105ffdb1
- candidate record SHA-256: d928c00a19602113202b7df1ea44c5653edd91c871d2d0638357b31a2fc4fed5
- semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06f
- question-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fc
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261012-r04-set-membership-0001.json)