Explore related knowledge and questions →
← Question derivation index · Original question batch
R0010-05671 · q-20261008-advance-20261012-r04-set-membership-0096
Finite integer set U=[-20, -1, 0, 1, 20]; P(x) means 20 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).
G001 — Finite integer set U=[-20, -1, 0, 1, 20]; P(x) means 20 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).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
declared finite type or bounded integer universe
After
membership truth is distinct from eligibility to ask the question
Conditions: none
Being outside the type is not the same as false membership.Check only the declared finite type or integer set,not all set theory or unrestricted comprehension.
Rule: finite_membership_scope · Uses: G001
Before
test 20 divides -20
After
-20=-1·20+0
Conditions: none
Euclidean division by positive |a| gives remainder0 exactly when the original nonzero a divides x.
Rule: membership_divisibility_witness · Uses: G001, S001
Before
x=-20;predicate=divides 20
After
True
Conditions: none
Evaluate every element in normalized U order;predicate success does not expand U.
Rule: finite_membership_predicate_row · Uses: G001, S002
Before
test 20 divides -1
After
-1=-1·20+19
Conditions: none
Euclidean division by positive |a| gives remainder0 exactly when the original nonzero a divides x.
Rule: membership_divisibility_witness · Uses: G001, S003
Before
x=-1;predicate=divides 20
After
False
Conditions: none
Evaluate every element in normalized U order;predicate success does not expand U.
Rule: finite_membership_predicate_row · Uses: G001, S004
Before
test 20 divides 0
After
0=0·20+0
Conditions: none
Euclidean division by positive |a| gives remainder0 exactly when the original nonzero a divides x.
Rule: membership_divisibility_witness · Uses: G001, S005
Before
x=0;predicate=divides 20
After
True
Conditions: none
Evaluate every element in normalized U order;predicate success does not expand U.
Rule: finite_membership_predicate_row · Uses: G001, S006
Before
test 20 divides 1
After
1=0·20+1
Conditions: none
Euclidean division by positive |a| gives remainder0 exactly when the original nonzero a divides x.
Rule: membership_divisibility_witness · Uses: G001, S007
Before
x=1;predicate=divides 20
After
False
Conditions: none
Evaluate every element in normalized U order;predicate success does not expand U.
Rule: finite_membership_predicate_row · Uses: G001, S008
Before
test 20 divides 20
After
20=1·20+0
Conditions: none
Euclidean division by positive |a| gives remainder0 exactly when the original nonzero a divides x.
Rule: membership_divisibility_witness · Uses: G001, S009
Before
x=20;predicate=divides 20
After
True
Conditions: none
Evaluate every element in normalized U order;predicate success does not expand U.
Rule: finite_membership_predicate_row · Uses: G001, S010
Before
normalized U=[-20, -1, 0, 1, 20];truths=[True, False, True, False, True]
After
A={x∈U:P(x)}=[-20, 0, 20]Conditions: none
The full extension contains exactly the successful elements of the supplied U;the truth list remains aligned with all sorted U-elements.Empty U gives two empty lists.
Rule: bounded_predicate_extension · Uses: G001, S011
S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.([-20, 0, 20], [True, False, True, False, True])
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: 851e644a67bd49409294934c4ead9bf5102932aa6d4309710cdc603280d08edecandidate record SHA-256: 965e3dba985a826422362105733b6d031726177c7e556775b0c03ebe656a9bc5semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06fquestion-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fcLean: NOT_RUN · human review: NOT_PERFORMED