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R0007-04421 · q-20261008-advance-20261010-r02-set-operations-0071
Finite integer sets A=[0] and B=[-1, 20]; repeated elements count once. The explicit universe is U=[-20, -1, 0, 1, 20], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
G001 — Finite integer sets A=[0] and B=[-1, 20]; repeated elements count once. The explicit universe is U=[-20, -1, 0, 1, 20], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.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=[0];raw B=[-1, 20]
After
A=[0];B=[-1, 20]
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=No;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=-1
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, S002
Before
element=0
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, S003
Before
element=1
After
in A=No;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, S004
Before
element=20
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, S005
Before
explicit U=[-20, -1, 0, 1, 20];U\A=[-20, -1, 1, 20];U\B=[-20, 0, 1]
After
U\(A∪B)=[-20, 1];(U\A)∩(U\B)=[-20, 1]
Conditions: none
Both expressions require membership in neither A nor B;complements are relative only to the declared U,not arbitrary outside integers.
Rule: relative_demorgan_complements · Uses: G001, S006
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 — 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, 1], [-20, 1])
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: 5cbc4863beb44c1b956d0b256fbddcfd16a417c0d61be4c147b8b1da38a3e996candidate record SHA-256: 3ebe7f103d1b9a9ddd498a60805af1312f7b69987526869ad5231cb2bdc52fbbsemantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485aquestion-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42Lean: NOT_RUN · human review: NOT_PERFORMED