This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261008-advance-20261010-r02-set-operations-foundations · 2026-10-08 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Finite integer sets A=[] and B=[]; repeated elements count once. Give (|A∪B|, |A∩B|).
(0, 0)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (0, 0).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[0] and B=[0]; repeated elements count once. Give (|A∪B|, |A∩B|).
(1, 1)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (1, 1).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[] and B=[100]; repeated elements count once. Give (|A∪B|, |A∩B|).
(1, 0)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (1, 0).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[0, 1] and B=[0, 1]; repeated elements count once. Give (|A∪B|, |A∩B|).
(2, 2)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (2, 2).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[0] and B=[0, 100]; repeated elements count once. Give (|A∪B|, |A∩B|).
(2, 1)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (2, 1).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[] and B=[100, 101]; repeated elements count once. Give (|A∪B|, |A∩B|).
(2, 0)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (2, 0).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[-1] and B=[100]; repeated elements count once. Give (|A∪B|, |A∩B|).
(2, 0)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (2, 0).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[0, 1, 2] and B=[0, 1, 2]; repeated elements count once. Give (|A∪B|, |A∩B|).
(3, 3)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (3, 3).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[0, 1] and B=[0, 1, 100]; repeated elements count once. Give (|A∪B|, |A∩B|).
(3, 2)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (3, 2).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[0] and B=[0, 100, 101]; repeated elements count once. Give (|A∪B|, |A∩B|).
(3, 1)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (3, 1).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[] and B=[100, 101, 102]; repeated elements count once. Give (|A∪B|, |A∩B|).
(3, 0)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (3, 0).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[0, -1] and B=[0, 100]; repeated elements count once. Give (|A∪B|, |A∩B|).
(3, 1)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (3, 1).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[-1] and B=[100, 101]; repeated elements count once. Give (|A∪B|, |A∩B|).
(3, 0)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (3, 0).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[0, 1, 2, 3] and B=[0, 1, 2, 3]; repeated elements count once. Give (|A∪B|, |A∩B|).
(4, 4)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (4, 4).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[0, 1, 2] and B=[0, 1, 2, 100]; repeated elements count once. Give (|A∪B|, |A∩B|).
(4, 3)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (4, 3).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[0, 1] and B=[0, 1, 100, 101]; repeated elements count once. Give (|A∪B|, |A∩B|).
(4, 2)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (4, 2).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[0] and B=[0, 100, 101, 102]; repeated elements count once. Give (|A∪B|, |A∩B|).
(4, 1)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (4, 1).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[] and B=[100, 101, 102, 103]; repeated elements count once. Give (|A∪B|, |A∩B|).
(4, 0)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (4, 0).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[0, 1, -1] and B=[0, 1, 100]; repeated elements count once. Give (|A∪B|, |A∩B|).
(4, 2)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (4, 2).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[0, -1] and B=[0, 100, 101]; repeated elements count once. Give (|A∪B|, |A∩B|).
(4, 1)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (4, 1).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[-1] and B=[100, 101, 102]; repeated elements count once. Give (|A∪B|, |A∩B|).
(4, 0)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (4, 0).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[-2, -1] and B=[100, 101]; repeated elements count once. Give (|A∪B|, |A∩B|).
(4, 0)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (4, 0).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[0, 1, 2, 3, 4] and B=[0, 1, 2, 3, 4]; repeated elements count once. Give (|A∪B|, |A∩B|).
(5, 5)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (5, 5).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[0, 1, 2, 3] and B=[0, 1, 2, 3, 100]; repeated elements count once. Give (|A∪B|, |A∩B|).
(5, 4)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (5, 4).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[0, 1, 2] and B=[0, 1, 2, 100, 101]; repeated elements count once. Give (|A∪B|, |A∩B|).
(5, 3)
Count membership in the two sets; common elements enter the union once, and the empty set has size zero. The exact result, retaining the requested order, is (5, 3).
Reviewed step-by-step derivation
Set operations · qf-density10-set-union-intersection-sizes · candidate
Finite integer sets A=[] and B=[]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[0, 1] and B=[0, 1]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[] and B=[-2, 2]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-2, 2]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-2, 2].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-3, 0, 3] and B=[0, 4]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-3, 3, 4]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-3, 3, 4].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-4, 0, 4] and B=[0, 5]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-4, 4, 5]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-4, 4, 5].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-5, 0, 5] and B=[0, 6]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-5, 5, 6]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-5, 5, 6].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-6, 0, 6] and B=[0, 7]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-6, 6, 7]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-6, 6, 7].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-7, 0, 7] and B=[0, 8]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-7, 7, 8]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-7, 7, 8].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-8, 0, 8] and B=[0, 9]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-8, 8, 9]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-8, 8, 9].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-9, 0, 9] and B=[0, 10]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-9, 9, 10]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-9, 9, 10].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-10, 0, 10] and B=[0, 11]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-10, 10, 11]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-10, 10, 11].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-11, 0, 11] and B=[0, 12]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-11, 11, 12]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-11, 11, 12].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-12, 0, 12] and B=[0, 13]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-12, 12, 13]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-12, 12, 13].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-13, 0, 13] and B=[0, 14]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-13, 13, 14]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-13, 13, 14].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-14, 0, 14] and B=[0, 15]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-14, 14, 15]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-14, 14, 15].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-15, 0, 15] and B=[0, 16]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-15, 15, 16]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-15, 15, 16].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-16, 0, 16] and B=[0, 17]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-16, 16, 17]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-16, 16, 17].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-17, 0, 17] and B=[0, 18]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-17, 17, 18]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-17, 17, 18].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-18, 0, 18] and B=[0, 19]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-18, 18, 19]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-18, 18, 19].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-19, 0, 19] and B=[0, 20]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-19, 19, 20]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-19, 19, 20].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-20, 0, 20] and B=[0, 21]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-20, 20, 21]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-20, 20, 21].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-21, 0, 21] and B=[0, 22]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-21, 21, 22]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-21, 21, 22].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-22, 0, 22] and B=[0, 23]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-22, 22, 23]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-22, 22, 23].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-23, 0, 23] and B=[0, 24]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-23, 23, 24]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-23, 23, 24].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[-24, 0, 24] and B=[0, 25]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.
[-24, 24, 25]
Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-24, 24, 25].
Reviewed step-by-step derivation
Set operations · qf-density10-set-symmetric-difference · candidate
Finite integer sets A=[] and B=[]; repeated elements count once. The explicit universe is U=[], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([], [])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([], []).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[-1, 0] and B=[1]; repeated elements count once. The explicit universe is U=[-1, 0, 1], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([], [])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([], []).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 2]; repeated elements count once. The explicit universe is U=[-2, -1, 0, 1, 2], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-2, 1], [-2, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-2, 1], [-2, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 3]; repeated elements count once. The explicit universe is U=[-3, -1, 0, 1, 3], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-3, 1], [-3, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-3, 1], [-3, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 4]; repeated elements count once. The explicit universe is U=[-4, -1, 0, 1, 4], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-4, 1], [-4, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-4, 1], [-4, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 5]; repeated elements count once. The explicit universe is U=[-5, -1, 0, 1, 5], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-5, 1], [-5, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-5, 1], [-5, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 6]; repeated elements count once. The explicit universe is U=[-6, -1, 0, 1, 6], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-6, 1], [-6, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-6, 1], [-6, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 7]; repeated elements count once. The explicit universe is U=[-7, -1, 0, 1, 7], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-7, 1], [-7, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-7, 1], [-7, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 8]; repeated elements count once. The explicit universe is U=[-8, -1, 0, 1, 8], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-8, 1], [-8, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-8, 1], [-8, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 9]; repeated elements count once. The explicit universe is U=[-9, -1, 0, 1, 9], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-9, 1], [-9, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-9, 1], [-9, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 10]; repeated elements count once. The explicit universe is U=[-10, -1, 0, 1, 10], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-10, 1], [-10, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-10, 1], [-10, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 11]; repeated elements count once. The explicit universe is U=[-11, -1, 0, 1, 11], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-11, 1], [-11, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-11, 1], [-11, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 12]; repeated elements count once. The explicit universe is U=[-12, -1, 0, 1, 12], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-12, 1], [-12, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-12, 1], [-12, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 13]; repeated elements count once. The explicit universe is U=[-13, -1, 0, 1, 13], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-13, 1], [-13, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-13, 1], [-13, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 14]; repeated elements count once. The explicit universe is U=[-14, -1, 0, 1, 14], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-14, 1], [-14, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-14, 1], [-14, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 15]; repeated elements count once. The explicit universe is U=[-15, -1, 0, 1, 15], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-15, 1], [-15, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-15, 1], [-15, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 16]; repeated elements count once. The explicit universe is U=[-16, -1, 0, 1, 16], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-16, 1], [-16, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-16, 1], [-16, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 17]; repeated elements count once. The explicit universe is U=[-17, -1, 0, 1, 17], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-17, 1], [-17, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-17, 1], [-17, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 18]; repeated elements count once. The explicit universe is U=[-18, -1, 0, 1, 18], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-18, 1], [-18, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-18, 1], [-18, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 19]; repeated elements count once. The explicit universe is U=[-19, -1, 0, 1, 19], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-19, 1], [-19, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-19, 1], [-19, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
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.
([-20, 1], [-20, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-20, 1], [-20, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 21]; repeated elements count once. The explicit universe is U=[-21, -1, 0, 1, 21], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-21, 1], [-21, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-21, 1], [-21, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 22]; repeated elements count once. The explicit universe is U=[-22, -1, 0, 1, 22], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-22, 1], [-22, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-22, 1], [-22, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 23]; repeated elements count once. The explicit universe is U=[-23, -1, 0, 1, 23], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-23, 1], [-23, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-23, 1], [-23, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[0] and B=[-1, 24]; repeated elements count once. The explicit universe is U=[-24, -1, 0, 1, 24], with A,B⊆U. Give (U\(A∪B), (U\A)∩(U\B)), each sorted increasingly.
([-24, 1], [-24, 1])
Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-24, 1], [-24, 1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-demorgan-sets · candidate
Finite integer sets A=[] and B=[]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([], [], [])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([], [], []).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-1, 0] and B=[0, 1]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-1], [0], [1])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-1], [0], [1]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-2, 0, 2] and B=[0, 2, 3]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-2], [0, 2], [3])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-2], [0, 2], [3]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-3, 0, 3] and B=[0, 3, 4]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-3], [0, 3], [4])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-3], [0, 3], [4]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-4, 0, 4] and B=[0, 4, 5]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-4], [0, 4], [5])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-4], [0, 4], [5]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
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.
([-5], [0, 5], [6])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-5], [0, 5], [6]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-6, 0, 6] and B=[0, 6, 7]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-6], [0, 6], [7])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-6], [0, 6], [7]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-7, 0, 7] and B=[0, 7, 8]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-7], [0, 7], [8])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-7], [0, 7], [8]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-8, 0, 8] and B=[0, 8, 9]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-8], [0, 8], [9])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-8], [0, 8], [9]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-9, 0, 9] and B=[0, 9, 10]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-9], [0, 9], [10])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-9], [0, 9], [10]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-10, 0, 10] and B=[0, 10, 11]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-10], [0, 10], [11])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-10], [0, 10], [11]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-11, 0, 11] and B=[0, 11, 12]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-11], [0, 11], [12])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-11], [0, 11], [12]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-12, 0, 12] and B=[0, 12, 13]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-12], [0, 12], [13])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-12], [0, 12], [13]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-13, 0, 13] and B=[0, 13, 14]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-13], [0, 13], [14])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-13], [0, 13], [14]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-14, 0, 14] and B=[0, 14, 15]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-14], [0, 14], [15])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-14], [0, 14], [15]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-15, 0, 15] and B=[0, 15, 16]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-15], [0, 15], [16])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-15], [0, 15], [16]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-16, 0, 16] and B=[0, 16, 17]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-16], [0, 16], [17])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-16], [0, 16], [17]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-17, 0, 17] and B=[0, 17, 18]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-17], [0, 17], [18])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-17], [0, 17], [18]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-18, 0, 18] and B=[0, 18, 19]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-18], [0, 18], [19])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-18], [0, 18], [19]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-19, 0, 19] and B=[0, 19, 20]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-19], [0, 19], [20])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-19], [0, 19], [20]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
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.
([-20], [0, 20], [21])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-20], [0, 20], [21]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-21, 0, 21] and B=[0, 21, 22]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-21], [0, 21], [22])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-21], [0, 21], [22]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-22, 0, 22] and B=[0, 22, 23]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-22], [0, 22], [23])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-22], [0, 22], [23]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-23, 0, 23] and B=[0, 23, 24]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-23], [0, 23], [24])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-23], [0, 23], [24]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Finite integer sets A=[-24, 0, 24] and B=[0, 24, 25]; repeated elements count once. Give (A\B, A∩B, B\A), each sorted increasingly, retaining empty parts.
([-24], [0, 24], [25])
Partition by left-only, common, and right-only membership without exchanging the two sides. The exact result, retaining the requested order, is ([-24], [0, 24], [25]).
Reviewed step-by-step derivation
Set operations · qf-density10-set-three-way-partition · candidate
Canonical question JSONL · English JSONL · Validation report