Question 4341: Negated Universal and Existential Pair

Original question and givens

The explicitly finite integer domain is U=[-5, -4, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.

• G001 — The explicitly finite integer domain is U=[-5, -4, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.

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=[-5, -4, -3, -2, -1]

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

x=-5;predicate=ge 0

P(x)=No

x=-4;predicate=ge 0

P(x)=No

x=-3;predicate=ge 0

P(x)=No

x=-2;predicate=ge 0

P(x)=No

x=-1;predicate=ge 0

P(x)=No

[[-5, No], [-4, No], [-3, No], [-2, No], [-1, No]]

T=[];F=[-5, -4, -3, -2, -1];T∪F=U and T∩F=∅

false-element count=5;true-element count=0

(True, True)

Reverse checks

• 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.

Result

(True, True)

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.