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R0007-04476 · q-20261008-advance-20261010-r02-set-operations-0001
Finite integer sets A=[] and B=[]; repeated elements count once. Give (|A∪B|, |A∩B|).
G001 — Finite integer sets A=[] and B=[]; repeated elements count once. Give (|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
raw A=[];raw B=[]
After
A=[];B=[]
Conditions: none
Deduplicate and numerically sort integer sets;repetition and input order do not change membership.The original lists remain retained.
Rule: finite_set_extensionality · Uses: G001
Before
separate set cardinalities sum := sum(0, 0)
After
separate set cardinalities sum = 0
Conditions: none
Adding the two set sizes counts their intersection twice.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
union cardinality check := sub(0, 0)
After
union cardinality check = 0
Conditions: none
Subtract one intersection count and compare with the deduplicated union size.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
union=[];intersection=[]
After
(0, 0)
Conditions: none
Report union size before intersection size;the empty set has size0 and a shared element is counted once in the union.
Rule: union_intersection_cardinality · Uses: G001, S003
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 — Recompute union cardinality check with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute separate set cardinalities sum with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(0, 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: fa619d978530132b084d7bc50f0152688f8d7972c72c391fad23005bfb6a6ad6candidate record SHA-256: 28a655c3dc469302716bfc95f03a34c129edbd2475913b6aadd7a8048521ea12semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485aquestion-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42Lean: NOT_RUN · human review: NOT_PERFORMED