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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-20261011-r05-subset-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=[1]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-1] and B=[]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-1]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-1].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-2, 2, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-2, 2]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-2, 2].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-3, 3, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-3, 3]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-3, 3].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-4, 4, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-4, 4]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-4, 4].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-5, 5, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-5, 5]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-5, 5].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-6, 6, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-6, 6]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-6, 6].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-7, 7, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-7, 7]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-7, 7].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-8, 8, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-8, 8]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-8, 8].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-9, 9, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-9, 9]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-9, 9].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-10, 10, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-10, 10]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-10, 10].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-11, 11, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-11, 11]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-11, 11].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-12, 12, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-12, 12]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-12, 12].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-13, 13, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-13, 13]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-13, 13].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-14, 14, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-14, 14]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-14, 14].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-15, 15, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-15, 15]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-15, 15].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-16, 16, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-16, 16]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-16, 16].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-17, 17, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-17, 17]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-17, 17].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-18, 18, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-18, 18]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-18, 18].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-19, 19, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-19, 19]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-19, 19].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-20, 20, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-20, 20]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-20, 20].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-21, 21, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-21, 21]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-21, 21].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-22, 22, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-22, 22]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-22, 22].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-23, 23, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-23, 23]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-23, 23].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[-24, 24, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.
[-24, 24]
The obstruction set is A\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-24, 24].
Reviewed step-by-step derivation
Subset · qf-density11-subset-obstruction-list · candidate
Finite integer sets A=[0] and B=[0]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(Yes, No, [], [])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (Yes, No, [], []).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[] and B=[-1]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(Yes, Yes, [], [-1])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (Yes, Yes, [], [-1]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[-2] and B=[]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(No, No, [-2], [])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (No, No, [-2], []).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[0] and B=[0, 3]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(Yes, Yes, [], [3])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (Yes, Yes, [], [3]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[-4, 0] and B=[0]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(No, No, [-4], [])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (No, No, [-4], []).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[-5, 0] and B=[0, 5]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(No, No, [-5], [5])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (No, No, [-5], [5]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[0] and B=[0, 6]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(Yes, Yes, [], [6])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (Yes, Yes, [], [6]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[-7, 0] and B=[0]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(No, No, [-7], [])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (No, No, [-7], []).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[-8, 0] and B=[0, 8]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(No, No, [-8], [8])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (No, No, [-8], [8]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[0] and B=[0, 9]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(Yes, Yes, [], [9])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (Yes, Yes, [], [9]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[-10, 0] and B=[0]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(No, No, [-10], [])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (No, No, [-10], []).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[-11, 0] and B=[0, 11]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(No, No, [-11], [11])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (No, No, [-11], [11]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[0] and B=[0, 12]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(Yes, Yes, [], [12])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (Yes, Yes, [], [12]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[-13, 0] and B=[0]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(No, No, [-13], [])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (No, No, [-13], []).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[-14, 0] and B=[0, 14]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(No, No, [-14], [14])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (No, No, [-14], [14]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[0] and B=[0, 15]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(Yes, Yes, [], [15])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (Yes, Yes, [], [15]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[-16, 0] and B=[0]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(No, No, [-16], [])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (No, No, [-16], []).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[-17, 0] and B=[0, 17]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(No, No, [-17], [17])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (No, No, [-17], [17]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[0] and B=[0, 18]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(Yes, Yes, [], [18])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (Yes, Yes, [], [18]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[-19, 0] and B=[0]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(No, No, [-19], [])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (No, No, [-19], []).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[-20, 0] and B=[0, 20]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(No, No, [-20], [20])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (No, No, [-20], [20]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[0] and B=[0, 21]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(Yes, Yes, [], [21])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (Yes, Yes, [], [21]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[-22, 0] and B=[0]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(No, No, [-22], [])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (No, No, [-22], []).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[-23, 0] and B=[0, 23]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(No, No, [-23], [23])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (No, No, [-23], [23]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[0] and B=[0, 24]; repeated elements count once. Give (does A⊆B?, does A⊊B?, A\B, B\A), with both witness sets increasing.
(Yes, Yes, [], [24])
An empty left difference permits inclusion; strict inclusion additionally requires a nonempty right difference. Equal sets pass only the non-strict field. The exact result, retaining the requested order, is (Yes, Yes, [], [24]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-strict-witnesses · candidate
Finite integer sets A=[] and B=[0]; repeated elements count once. C=[0, 1] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([], [], [])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([], [], []).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[1] and B=[]; repeated elements count once. C=[1] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([1], [], [])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([1], [], []).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[0] and B=[0, 2]; repeated elements count once. C=[0] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([], [2], [])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([], [2], []).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-3, 0] and B=[0, 3]; repeated elements count once. C=[0, 4] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-3], [3], [-3])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-3], [3], [-3]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-4, 0] and B=[0, 4]; repeated elements count once. C=[0, 5] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-4], [4], [-4])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-4], [4], [-4]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-5, 0] and B=[0, 5]; repeated elements count once. C=[0, 6] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-5], [5], [-5])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-5], [5], [-5]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-6, 0] and B=[0, 6]; repeated elements count once. C=[0, 7] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-6], [6], [-6])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-6], [6], [-6]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-7, 0] and B=[0, 7]; repeated elements count once. C=[0, 8] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-7], [7], [-7])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-7], [7], [-7]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-8, 0] and B=[0, 8]; repeated elements count once. C=[0, 9] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-8], [8], [-8])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-8], [8], [-8]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-9, 0] and B=[0, 9]; repeated elements count once. C=[0, 10] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-9], [9], [-9])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-9], [9], [-9]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-10, 0] and B=[0, 10]; repeated elements count once. C=[0, 11] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-10], [10], [-10])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-10], [10], [-10]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-11, 0] and B=[0, 11]; repeated elements count once. C=[0, 12] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-11], [11], [-11])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-11], [11], [-11]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-12, 0] and B=[0, 12]; repeated elements count once. C=[0, 13] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-12], [12], [-12])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-12], [12], [-12]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-13, 0] and B=[0, 13]; repeated elements count once. C=[0, 14] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-13], [13], [-13])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-13], [13], [-13]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-14, 0] and B=[0, 14]; repeated elements count once. C=[0, 15] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-14], [14], [-14])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-14], [14], [-14]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-15, 0] and B=[0, 15]; repeated elements count once. C=[0, 16] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-15], [15], [-15])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-15], [15], [-15]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-16, 0] and B=[0, 16]; repeated elements count once. C=[0, 17] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-16], [16], [-16])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-16], [16], [-16]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-17, 0] and B=[0, 17]; repeated elements count once. C=[0, 18] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-17], [17], [-17])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-17], [17], [-17]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-18, 0] and B=[0, 18]; repeated elements count once. C=[0, 19] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-18], [18], [-18])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-18], [18], [-18]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-19, 0] and B=[0, 19]; repeated elements count once. C=[0, 20] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-19], [19], [-19])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-19], [19], [-19]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-20, 0] and B=[0, 20]; repeated elements count once. C=[0, 21] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-20], [20], [-20])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-20], [20], [-20]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-21, 0] and B=[0, 21]; repeated elements count once. C=[0, 22] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-21], [21], [-21])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-21], [21], [-21]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-22, 0] and B=[0, 22]; repeated elements count once. C=[0, 23] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-22], [22], [-22])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-22], [22], [-22]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-23, 0] and B=[0, 23]; repeated elements count once. C=[0, 24] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-23], [23], [-23])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-23], [23], [-23]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[-24, 0] and B=[0, 24]; repeated elements count once. C=[0, 25] uses the same integer type. Give (A\B, B\C, A\C), each increasing, retaining empty parts.
([-24], [24], [-24])
Enumerate each difference independently; a failed link does not automatically decide another difference, and the three fields are not interchangeable. The exact result, retaining the requested order, is ([-24], [24], [-24]).
Reviewed step-by-step derivation
Subset · qf-density11-subset-chain-obstructions · candidate
Finite integer sets A=[] and B=[]; repeated elements count once. Ordered candidates=[100, 101, 102]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[No, No, No]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[-1, 101, 102]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[Yes, No, No]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[1, 2, 102]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[No, No, No]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[-1, 2, 102]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[Yes, No, No]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[100, -2, 102]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[No, Yes, No]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [No, Yes, No].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[1, -2, 102]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[No, Yes, No]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [No, Yes, No].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[-1, -2, 102]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[Yes, Yes, No]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [Yes, Yes, No].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[100, 101, 3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[No, No, No]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[1, 101, 3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[No, No, No]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[-1, 101, 3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[Yes, No, No]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[100, 2, 3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[No, No, No]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[1, 2, 3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[No, No, No]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[-1, 2, 3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[Yes, No, No]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[100, -2, 3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[No, Yes, No]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [No, Yes, No].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[1, -2, 3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[No, Yes, No]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [No, Yes, No].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[-1, -2, 3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[Yes, Yes, No]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [Yes, Yes, No].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[100, 101, -3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[No, No, Yes]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [No, No, Yes].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[1, 101, -3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[No, No, Yes]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [No, No, Yes].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[-1, 101, -3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[Yes, No, Yes]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [Yes, No, Yes].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[100, 2, -3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[No, No, Yes]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [No, No, Yes].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[1, 2, -3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[No, No, Yes]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [No, No, Yes].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[-1, 2, -3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[Yes, No, Yes]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [Yes, No, Yes].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[100, -2, -3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[No, Yes, Yes]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [No, Yes, Yes].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[1, -2, -3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[No, Yes, Yes]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [No, Yes, Yes].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[]; repeated elements count once. Ordered candidates=[-1, -2, -3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.
[Yes, Yes, Yes]
Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [Yes, Yes, Yes].
Reviewed step-by-step derivation
Subset · qf-density11-subset-candidate-witness-mask · candidate
Canonical question JSONL · English JSONL · Validation report