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R0009-05378 · q-20261008-advance-20261012-r02-counting-principle-0078
Finite integer choice sets A=[], B=[100]. Give (|A|+|B|, |A∪B|, |A∩B|).
G001 — Finite integer choice sets A=[], B=[100]. Give (|A|+|B|, |A∪B|, |A∩B|).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.
Before
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.
Rule: finite_counting_protocol_scope · Uses: G001
Before
choice=100;in A=No;in B=Yes
After
shared=No
Conditions: none
A shared element appears twice in the sum of sizes but once in the union.
Rule: counting_overlap_membership · Uses: G001, S001
Before
sum of choice-set sizes := sum(0, 1)
After
sum of choice-set sizes = 1
Conditions: none
First add the two set sizes,which may double-count shared choices.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
overlap-corrected union size := sub(1, 0)
After
overlap-corrected union size = 1
Conditions: none
Subtract the intersection once and audit the actual union size.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
A∪B=[100];A∩B=[]
After
union size=1
Conditions: none
Without disjoint tags,apply the overlap correction rather than treating one shared choice as two outcomes.
Rule: finite_choice_inclusion_exclusion · Uses: G001, S004
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 — Recompute overlap-corrected union size with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute sum of choice-set sizes with exact rational arithmetic, not rounded decimals.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.(1, 1, 0)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: e6d74a76d0f8a73a673fd45f4840272fa93fe6fcef08038439439684f4d74043candidate record SHA-256: 9a8b24048a48b32e6c71ad44588e437faef7a5dd215833f7d84c72d96eaead08semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462Lean: NOT_RUN · human review: NOT_PERFORMED