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Question 4355: Two Ordered Nested-Quantifier Counts

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0007-04355 · q-20261008-advance-20261010-r07-quantifier-0080

Original question and givens

U=[-2, -1, 0, 1, 2], V=[-1, 0, 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=[-2, -1, 0, 1, 2], V=[-1, 0, 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.

Step-by-step derivation

S001 · implies

Before

complete finite U=[-2, -1, 0, 1, 2]

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=-1;relation=le

After

Yes

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=0;relation=le

After

Yes

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;y=1;relation=le

After

Yes

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependencies

Rule: finite_relation_cell · Uses: G001, S003

S005 · implies

Before

x=-2;row truths=[Yes, Yes, Yes]

After

some y satisfies=Yes

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, S004

S006 · implies

Before

x=-1;y=-1;relation=le

After

Yes

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;y=0;relation=le

After

Yes

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependencies

Rule: finite_relation_cell · Uses: G001, S006

S008 · implies

Before

x=-1;y=1;relation=le

After

Yes

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependencies

Rule: finite_relation_cell · Uses: G001, S007

S009 · implies

Before

x=-1;row truths=[Yes, Yes, Yes]

After

some y satisfies=Yes

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, S008

S010 · implies

Before

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

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependencies

Rule: finite_relation_cell · Uses: G001, S009

S011 · implies

Before

x=0;y=0;relation=le

After

Yes

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependencies

Rule: finite_relation_cell · Uses: G001, S010

S012 · implies

Before

x=0;y=1;relation=le

After

Yes

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependencies

Rule: finite_relation_cell · Uses: G001, S011

S013 · implies

Before

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

After

some y satisfies=Yes

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, S012

S014 · implies

Before

x=1;y=-1;relation=le

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependencies

Rule: finite_relation_cell · Uses: G001, S013

S015 · implies

Before

x=1;y=0;relation=le

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependencies

Rule: finite_relation_cell · Uses: G001, S014

S016 · implies

Before

x=1;y=1;relation=le

After

Yes

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependencies

Rule: finite_relation_cell · Uses: G001, S015

S017 · implies

Before

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

After

some y satisfies=Yes

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, S016

S018 · implies

Before

x=2;y=-1;relation=le

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependencies

Rule: finite_relation_cell · Uses: G001, S017

S019 · implies

Before

x=2;y=0;relation=le

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependencies

Rule: finite_relation_cell · Uses: G001, S018

S020 · implies

Before

x=2;y=1;relation=le

After

No

Conditions: none

Evaluate this ordered pair without exchanging x and y.

Step identifiers and dependencies

Rule: finite_relation_cell · Uses: G001, S019

S021 · implies

Before

x=2;row truths=[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 dependencies

Rule: nested_exists_row · Uses: G001, S020

S022 · implies

Before

y=-1;column truths=[Yes, Yes, No, 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, S021

S023 · implies

Before

y=0;column truths=[Yes, Yes, Yes, 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, S022

S024 · implies

Before

y=1;column truths=[Yes, Yes, Yes, Yes, 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, S023

S025 · implies

Before

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

After

(4, 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, S024

Reverse checks

  • S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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

(4, 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: e3f63e025c4c041b7be23ceec3df88f5deca49d8fcd8df14a0f457b5f07e5fa6
  • candidate record SHA-256: 6c4051d201b5a952ebcbf57dc8f28f12de417c6c966dd2dc9328de00082f23d0
  • semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a
  • question-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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