Question 4373: Two Ordered Nested-Quantifier Counts

Original question and givens

U=[0, -1], V=[0, 100, 101, 102] 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=[0, -1], V=[0, 100, 101, 102] 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=[-1, 0]

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

x=-1;y=0;relation=eq

No

x=-1;y=100;relation=eq

No

x=-1;y=101;relation=eq

No

x=-1;y=102;relation=eq

No

x=-1;row truths=[No, No, No, No]

some y satisfies=No

x=0;y=0;relation=eq

Yes

x=0;y=100;relation=eq

No

x=0;y=101;relation=eq

No

x=0;y=102;relation=eq

No

x=0;row truths=[Yes, No, No, No]

some y satisfies=Yes

y=0;column truths=[No, Yes]

all x satisfy=No

y=100;column truths=[No, No]

all x satisfy=No

y=101;column truths=[No, No]

all x satisfy=No

y=102;column truths=[No, No]

all x satisfy=No

exists-by-x=[No, Yes];forall-by-y=[No, No, No, No]

(1, 0)

Reverse checks

• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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.

Result

(1, 0)

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.