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R0007-04456 · q-20261008-advance-20261010-r02-set-operations-0081
Finite integer sets A=[-5, 0, 5] and B=[0, 5, 6]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
G001 — Finite integer sets A=[-5, 0, 5] and B=[0, 5, 6]; 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=[-5, 0, 5];raw B=[0, 5, 6]
After
A=[-5, 0, 5];B=[0, 5, 6]
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=-5
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=5
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=6
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=[-5, 0, 5, 6]
After
([-5], [0, 5], [6])
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.([-5], [0, 5], [6])
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: 8a08ce45ed18dad2ec3d0dd632ec0f75e3d265fe3fb0a8596fa95970d0fd713ccandidate record SHA-256: 9184f71c9240efb5f0c3edc8a23863814e0c098d88c52de4ce28b72c92e62bc1semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485aquestion-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42Lean: NOT_RUN · human review: NOT_PERFORMED