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R0007-04471 · q-20261008-advance-20261010-r02-set-operations-0096
Finite integer sets A=[-20, 0, 20] and B=[0, 20, 21]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
G001 — Finite integer sets A=[-20, 0, 20] and B=[0, 20, 21]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.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=[-20, 0, 20];raw B=[0, 20, 21]
After
A=[-20, 0, 20];B=[0, 20, 21]
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
element=-20
After
in A=Yes;in B=No
Conditions: none
Record membership on both sides for each element;these Boolean values determine the set operations without duplicate counting.
Rule: finite_set_membership_row · Uses: G001, S001
Before
element=0
After
in A=Yes;in B=Yes
Conditions: none
Record membership on both sides for each element;these Boolean values determine the set operations without duplicate counting.
Rule: finite_set_membership_row · Uses: G001, S002
Before
element=20
After
in A=Yes;in B=Yes
Conditions: none
Record membership on both sides for each element;these Boolean values determine the set operations without duplicate counting.
Rule: finite_set_membership_row · Uses: G001, S003
Before
element=21
After
in A=No;in B=Yes
Conditions: none
Record membership on both sides for each element;these Boolean values determine the set operations without duplicate counting.
Rule: finite_set_membership_row · Uses: G001, S004
Before
A∪B=[-20, 0, 20, 21]
After
([-20], [0, 20], [21])
Conditions: none
Keep the order left-only,shared,right-only;the parts are pairwise disjoint and cover the union,including empty parts.
Rule: ordered_three_way_set_partition · Uses: G001, S005
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.([-20], [0, 20], [21])
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: 1c2cbdb694b63817b8c22f9c24b52eb69099f02a665f69fdd1e7cdce6157ce3acandidate record SHA-256: 6aae8b4c8c928033c6c5489f52b33387d8b1095c6dbfbb72efd01fce5ccdc89fsemantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485aquestion-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42Lean: NOT_RUN · human review: NOT_PERFORMED