Question 4366: Two Ordered Nested-Quantifier Counts

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0007-04366 · q-20261008-advance-20261010-r07-quantifier-0091

Original question and givens

U=[-2, -1], V=[100, 101] 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] 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

S001 · implies

Before

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 dependencies

Rule: finite_quantifier_scope · Uses: G001

S002 · implies

Before

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

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependencies

Rule: finite_relation_cell · Uses: G001, S001

S003 · implies

Before

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

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependencies

Rule: finite_relation_cell · Uses: G001, S002

S004 · implies

Before

x=-2;row truths=[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 dependencies

Rule: nested_exists_row · Uses: G001, S003

S005 · implies

Before

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

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependencies

Rule: finite_relation_cell · Uses: G001, S004

S006 · implies

Before

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

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependencies

Rule: finite_relation_cell · Uses: G001, S005

S007 · implies

Before

x=-1;row truths=[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 dependencies

Rule: nested_exists_row · Uses: G001, S006

S008 · implies

Before

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 dependencies

Rule: nested_forall_column · Uses: G001, S007

S009 · implies

Before

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 dependencies

Rule: nested_forall_column · Uses: G001, S008

S010 · implies

Before

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

Rule: nested_quantifier_count_order · Uses: G001, S009

Reverse checks

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: 4cc6b6cc2f13343e6466381ffc3a899d88e2d61bf59eb5fdd445876c0aaff5df
candidate record SHA-256: 10810869927dba46ae221a3383a93becedae03407fc8cc6767ebddb19cab4e6c
semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a
question-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42
Lean: NOT_RUN · human review: NOT_PERFORMED

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