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R0009-05046 · q-20261008-advance-20261011-r04-set-0046
Finite integer sets A=[-20, 0, 20] and B=[0, 20, 21]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.
G001 — Finite integer sets A=[-20, 0, 20] and B=[0, 20, 21]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.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 integer list=[-20, 0, 20]
After
extensional set=[-20, 0, 20]
Conditions: none
A set depends on its elements;deduplication and numeric ordering do not change membership.The original list is retained as source.
Rule: set_extensionality_normalization · Uses: G001
Before
element=-20;in A=Yes;in B=No
After
in A\B=Yes;in B\A=No
Conditions: none
Check both directional differences;equal cardinalities alone do not imply equal elements.
Rule: two_sided_set_difference_row · Uses: G001, S001
Before
element=0;in A=Yes;in B=Yes
After
in A\B=No;in B\A=No
Conditions: none
Check both directional differences;equal cardinalities alone do not imply equal elements.
Rule: two_sided_set_difference_row · Uses: G001, S002
Before
element=20;in A=Yes;in B=Yes
After
in A\B=No;in B\A=No
Conditions: none
Check both directional differences;equal cardinalities alone do not imply equal elements.
Rule: two_sided_set_difference_row · Uses: G001, S003
Before
element=21;in A=No;in B=Yes
After
in A\B=No;in B\A=Yes
Conditions: none
Check both directional differences;equal cardinalities alone do not imply equal elements.
Rule: two_sided_set_difference_row · Uses: G001, S004
Before
A\B=[-20];B\A=[21]
After
sets equal=No
Conditions: none
Both differences must be empty to establish A⊆B and B⊆A,and thus extensional equality.One empty difference alone is insufficient.
Rule: set_equality_two_inclusions · Uses: G001, S005
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(No, [-20], [21])
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: df28c2a862e0f5f65fbd24e34858d50cad4b8350609bcd8e829a89914f699e5dcandidate record SHA-256: 5b8f691d5bef5440a0ec7420dc9ac96819003fc7b430e5eca887248f588d582dsemantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462Lean: NOT_RUN · human review: NOT_PERFORMED