Question 4387: Complete Finite Predicate Partition

Original question and givens

The explicitly finite integer domain is U=[-11, -1, 0, 1, 11] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.

• G001 — The explicitly finite integer domain is U=[-11, -1, 0, 1, 11] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.

Definition and scope

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.

• mko-quantifier

Step-by-step derivation

complete finite U=[-11, -1, 0, 1, 11]

universal on empty domain=True;existential on empty domain=False

x=-11;predicate=divides 2

P(x)=No

x=-1;predicate=divides 2

P(x)=No

x=0;predicate=divides 2

P(x)=Yes

x=1;predicate=divides 2

P(x)=No

x=11;predicate=divides 2

P(x)=No

[[-11, No], [-1, No], [0, Yes], [1, No], [11, No]]

T=[0];F=[-11, -1, 1, 11];T∪F=U and T∩F=∅

Reverse checks

• 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.

Result

([0], [-11, -1, 1, 11])

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

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.