# Question 4372: Two Ordered Nested-Quantifier Counts

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0007-04372 · q-20261008-advance-20261010-r07-quantifier-0097



## Original question and givens

U=[-2, -1], V=[100, 101, 102, 103] 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=[-2, -1], V=[100, 101, 102, 103] 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](https://ocme.evemisslab.com/en/mko/mko-quantifier/)





## Step-by-step derivation

S001 · impliesBefore

complete finite U=[-2, -1]

After

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

Conditions: none

Enumerate only the declared finite domain;empty-domain universals are true and existentials false.Complete finite enumeration does not extend to an infinite domain.

Step identifiers and dependenciesRule: finite_quantifier_scope · Uses: G001





S002 · impliesBefore

x=-2;y=100;relation=eq

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependenciesRule: finite_relation_cell · Uses: G001, S001





S003 · impliesBefore

x=-2;y=101;relation=eq

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependenciesRule: finite_relation_cell · Uses: G001, S002





S004 · impliesBefore

x=-2;y=102;relation=eq

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependenciesRule: finite_relation_cell · Uses: G001, S003





S005 · impliesBefore

x=-2;y=103;relation=eq

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependenciesRule: finite_relation_cell · Uses: G001, S004





S006 · impliesBefore

x=-2;row truths=[No, No, No, No]

After

some y satisfies=No

Conditions: none

The inner existential needs at least one true entry;when V is empty this condition is false for every x.

Step identifiers and dependenciesRule: nested_exists_row · Uses: G001, S005





S007 · impliesBefore

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

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependenciesRule: finite_relation_cell · Uses: G001, S006





S008 · impliesBefore

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

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependenciesRule: finite_relation_cell · Uses: G001, S007





S009 · impliesBefore

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

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependenciesRule: finite_relation_cell · Uses: G001, S008





S010 · impliesBefore

x=-1;y=103;relation=eq

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependenciesRule: finite_relation_cell · Uses: G001, S009





S011 · impliesBefore

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

After

some y satisfies=No

Conditions: none

The inner existential needs at least one true entry;when V is empty this condition is false for every x.

Step identifiers and dependenciesRule: nested_exists_row · Uses: G001, S010





S012 · impliesBefore

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

After

all x satisfy=No

Conditions: none

The inner universal requires every entry true;when U is empty it holds for each y.

Step identifiers and dependenciesRule: nested_forall_column · Uses: G001, S011





S013 · impliesBefore

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

After

all x satisfy=No

Conditions: none

The inner universal requires every entry true;when U is empty it holds for each y.

Step identifiers and dependenciesRule: nested_forall_column · Uses: G001, S012





S014 · impliesBefore

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

After

all x satisfy=No

Conditions: none

The inner universal requires every entry true;when U is empty it holds for each y.

Step identifiers and dependenciesRule: nested_forall_column · Uses: G001, S013





S015 · impliesBefore

y=103;column truths=[No, No]

After

all x satisfy=No

Conditions: none

The inner universal requires every entry true;when U is empty it holds for each y.

Step identifiers and dependenciesRule: nested_forall_column · Uses: G001, S014





S016 · impliesBefore

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

After

(0, 0)

Conditions: none

The first count concerns x with some suitable y;the second concerns y satisfying every x.These are distinct ordered quantified conditions.

Step identifiers and dependenciesRule: nested_quantifier_count_order · Uses: G001, S015









## 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

(0, 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: a25ce55b5652cbf87f48bb15366ed5c89a0ee483da363076b5a1f0615fbbf279
- candidate record SHA-256: 8e20f27222473bc591a7fd9041ed6d85c87aef26e809ee1cafc3ba31ce37b53a
- semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a
- question-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261010-r07-quantifier-0097.json)