Question 4384: Complete Finite Predicate Partition

Original question and givens

The explicitly finite integer domain is U=[-8, -1, 0, 1, 8] 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=[-8, -1, 0, 1, 8] 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=[-8, -1, 0, 1, 8]

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

x=-8;predicate=divides 2

P(x)=Yes

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=8;predicate=divides 2

P(x)=Yes

[[-8, Yes], [-1, No], [0, Yes], [1, No], [8, Yes]]

T=[-8, 0, 8];F=[-1, 1];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

([-8, 0, 8], [-1, 1])

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.