Question 4352: Two Ordered Nested-Quantifier Counts

Original question and givens

U=[], V=[-1, 1] are explicitly finite integer domains. P(x,y) means x≤y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).

• G001 — U=[], V=[-1, 1] are explicitly finite integer domains. P(x,y) means x≤y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).

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=[]

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

y=-1;column truths=[]

all x satisfy=Yes

y=1;column truths=[]

all x satisfy=Yes

exists-by-x=[];forall-by-y=[Yes, Yes]

(0, 2)

Reverse checks

• 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, 2)

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.