# Question 5383: Overlap-Corrected Finite Choice Counts

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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.

R0009-05383 · q-20261008-advance-20261012-r02-counting-principle-0083



## Original question and givens

Finite integer choice sets A=[0, 1, 2], B=[0, 1, 2]. Give (|A|+|B|, |A∪B|, |A∩B|).


- G001 — Finite integer choice sets A=[0, 1, 2], B=[0, 1, 2]. Give (|A|+|B|, |A∪B|, |A∩B|).





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





## Step-by-step derivation

S001 · impliesBefore

declared finite tagged choices or full-product protocol

After

combinatorial counts,not probabilities or independence

Conditions: none

Addition uses disjoint tags and multiplication uses the declared full cross-stage protocol;neither is a probability-independence assumption.

Step identifiers and dependenciesRule: finite_counting_protocol_scope · Uses: G001





S002 · impliesBefore

choice=0;in A=Yes;in B=Yes

After

shared=Yes

Conditions: none

A shared element appears twice in the sum of sizes but once in the union.

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





S003 · impliesBefore

choice=1;in A=Yes;in B=Yes

After

shared=Yes

Conditions: none

A shared element appears twice in the sum of sizes but once in the union.

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





S004 · impliesBefore

choice=2;in A=Yes;in B=Yes

After

shared=Yes

Conditions: none

A shared element appears twice in the sum of sizes but once in the union.

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





S005 · calculationBefore

sum of choice-set sizes := sum(3, 3)

After

sum of choice-set sizes = 6

Conditions: none

First add the two set sizes,which may double-count shared choices.

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





S006 · calculationBefore

overlap-corrected union size := sub(6, 3)

After

overlap-corrected union size = 3

Conditions: none

Subtract the intersection once and audit the actual union size.

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





S007 · impliesBefore

A∪B=[0, 1, 2];A∩B=[0, 1, 2]

After

union size=3

Conditions: none

Without disjoint tags,apply the overlap correction rather than treating one shared choice as two outcomes.

Step identifiers and dependenciesRule: finite_choice_inclusion_exclusion · Uses: G001, S006









## Reverse checks


- 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 — Recompute overlap-corrected union size with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute sum of choice-set sizes with exact rational arithmetic, not rounded decimals.
- 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

(6, 3, 3)

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: 127957daaf6b9dca18da21d68368faa6ce8e4a6402a9f4b775cc514f6a6ca144
- candidate record SHA-256: d0c4eb031e8fc6afe78f2cb8c85b266188382d2efa5dc969d8b208cfa1e0289b
- 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-20261012-r02-counting-principle-0083.json)