This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20260913-foundational-language · 2026-09-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Under this OCME entry's convention that N includes 0, does 0 belong to N?
Yes
0 is a nonnegative integer, so it belongs to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does 1 belong to N?
Yes
1 is a nonnegative integer, so it belongs to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does 2 belong to N?
Yes
2 is a nonnegative integer, so it belongs to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does 5 belong to N?
Yes
5 is a nonnegative integer, so it belongs to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does 10 belong to N?
Yes
10 is a nonnegative integer, so it belongs to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does 25 belong to N?
Yes
25 is a nonnegative integer, so it belongs to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does 100 belong to N?
Yes
100 is a nonnegative integer, so it belongs to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does 7 belong to N?
Yes
7 is a nonnegative integer, so it belongs to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does 3 belong to N?
Yes
3 is a nonnegative integer, so it belongs to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does 12 belong to N?
Yes
12 is a nonnegative integer, so it belongs to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does -1 belong to N?
No
-1 is not a nonnegative integer, so it does not belong to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does -2 belong to N?
No
-2 is not a nonnegative integer, so it does not belong to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does -10 belong to N?
No
-10 is not a nonnegative integer, so it does not belong to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does -25 belong to N?
No
-25 is not a nonnegative integer, so it does not belong to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does -100 belong to N?
No
-100 is not a nonnegative integer, so it does not belong to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does -0.5 belong to N?
No
-0.5 is not a nonnegative integer, so it does not belong to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does 0.5 belong to N?
No
0.5 is not a nonnegative integer, so it does not belong to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does 1.5 belong to N?
No
1.5 is not a nonnegative integer, so it does not belong to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does 2.25 belong to N?
No
2.25 is not a nonnegative integer, so it does not belong to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Under this OCME entry's convention that N includes 0, does -3.75 belong to N?
No
-3.75 is not a nonnegative integer, so it does not belong to N under this entry's convention.
Reviewed step-by-step derivation
Natural number · qf-natural-number-membership · candidate
Let A={1, 2} be an explicitly enumerated finite set. Does 2 satisfy 2∈A?
Yes
2 occurs in the explicit enumeration of A, so 2∈A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={11, 12, 13} be an explicitly enumerated finite set. Does 18 satisfy 18∈A?
No
18 does not occur in the explicit enumeration of A, so 18∉A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={21, 22, 23, 24} be an explicitly enumerated finite set. Does 24 satisfy 24∈A?
Yes
24 occurs in the explicit enumeration of A, so 24∈A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={31, 32} be an explicitly enumerated finite set. Does 37 satisfy 37∈A?
No
37 does not occur in the explicit enumeration of A, so 37∉A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={41, 42, 43} be an explicitly enumerated finite set. Does 43 satisfy 43∈A?
Yes
43 occurs in the explicit enumeration of A, so 43∈A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={51, 52, 53, 54} be an explicitly enumerated finite set. Does 59 satisfy 59∈A?
No
59 does not occur in the explicit enumeration of A, so 59∉A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={61, 62} be an explicitly enumerated finite set. Does 62 satisfy 62∈A?
Yes
62 occurs in the explicit enumeration of A, so 62∈A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={71, 72, 73} be an explicitly enumerated finite set. Does 78 satisfy 78∈A?
No
78 does not occur in the explicit enumeration of A, so 78∉A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={81, 82, 83, 84} be an explicitly enumerated finite set. Does 82 satisfy 82∈A?
Yes
82 occurs in the explicit enumeration of A, so 82∈A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={91, 92} be an explicitly enumerated finite set. Does 97 satisfy 97∈A?
No
97 does not occur in the explicit enumeration of A, so 97∉A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={101, 102, 103} be an explicitly enumerated finite set. Does 103 satisfy 103∈A?
Yes
103 occurs in the explicit enumeration of A, so 103∈A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={111, 112, 113, 114} be an explicitly enumerated finite set. Does 119 satisfy 119∈A?
No
119 does not occur in the explicit enumeration of A, so 119∉A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={121, 122} be an explicitly enumerated finite set. Does 122 satisfy 122∈A?
Yes
122 occurs in the explicit enumeration of A, so 122∈A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={131, 132, 133} be an explicitly enumerated finite set. Does 138 satisfy 138∈A?
No
138 does not occur in the explicit enumeration of A, so 138∉A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={141, 142, 143, 144} be an explicitly enumerated finite set. Does 144 satisfy 144∈A?
Yes
144 occurs in the explicit enumeration of A, so 144∈A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={151, 152} be an explicitly enumerated finite set. Does 157 satisfy 157∈A?
No
157 does not occur in the explicit enumeration of A, so 157∉A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={161, 162, 163} be an explicitly enumerated finite set. Does 163 satisfy 163∈A?
Yes
163 occurs in the explicit enumeration of A, so 163∈A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={171, 172, 173, 174} be an explicitly enumerated finite set. Does 179 satisfy 179∈A?
No
179 does not occur in the explicit enumeration of A, so 179∉A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={181, 182} be an explicitly enumerated finite set. Does 182 satisfy 182∈A?
Yes
182 occurs in the explicit enumeration of A, so 182∈A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={191, 192, 193} be an explicitly enumerated finite set. Does 198 satisfy 198∈A?
No
198 does not occur in the explicit enumeration of A, so 198∉A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={201, 202, 203, 204} be an explicitly enumerated finite set. Does 202 satisfy 202∈A?
Yes
202 occurs in the explicit enumeration of A, so 202∈A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={211, 212} be an explicitly enumerated finite set. Does 217 satisfy 217∈A?
No
217 does not occur in the explicit enumeration of A, so 217∉A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={221, 222, 223} be an explicitly enumerated finite set. Does 223 satisfy 223∈A?
Yes
223 occurs in the explicit enumeration of A, so 223∈A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={231, 232, 233, 234} be an explicitly enumerated finite set. Does 239 satisfy 239∈A?
No
239 does not occur in the explicit enumeration of A, so 239∉A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
Let A={241, 242} be an explicitly enumerated finite set. Does 242 satisfy 242∈A?
Yes
242 occurs in the explicit enumeration of A, so 242∈A.
Reviewed step-by-step derivation
Set membership · qf-finite-set-membership · candidate
In the introductory classical-logic context specified here, is “0 belongs to the N used by this entry.” a proposition? Classify it only; do not decide its truth or proof status.
Yes
It is a closed statement whose truth value can be discussed in the specified classical context. Being a proposition does not mean it has been proved true.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “7=7.” a proposition? Classify it only; do not decide its truth or proof status.
Yes
It is a closed statement whose truth value can be discussed in the specified classical context. Being a proposition does not mean it has been proved true.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “3 is greater than 9.” a proposition? Classify it only; do not decide its truth or proof status.
Yes
It is a closed statement whose truth value can be discussed in the specified classical context. Being a proposition does not mean it has been proved true.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “0=1.” a proposition? Classify it only; do not decide its truth or proof status.
Yes
It is a closed statement whose truth value can be discussed in the specified classical context. Being a proposition does not mean it has been proved true.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “5 is less than 8.” a proposition? Classify it only; do not decide its truth or proof status.
Yes
It is a closed statement whose truth value can be discussed in the specified classical context. Being a proposition does not mean it has been proved true.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “x>0.” a proposition? Classify it only; do not decide its truth or proof status.
No
It contains an unspecified free variable and is therefore an open sentence. It requires assignment, substitution, or quantification before evaluation.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “n is even.” a proposition? Classify it only; do not decide its truth or proof status.
No
It contains an unspecified free variable and is therefore an open sentence. It requires assignment, substitution, or quantification before evaluation.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “y∈A.” a proposition? Classify it only; do not decide its truth or proof status.
No
It contains an unspecified free variable and is therefore an open sentence. It requires assignment, substitution, or quantification before evaluation.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “a+b=c.” a proposition? Classify it only; do not decide its truth or proof status.
No
It contains an unspecified free variable and is therefore an open sentence. It requires assignment, substitution, or quantification before evaluation.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “f(x)=y.” a proposition? Classify it only; do not decide its truth or proof status.
No
It contains an unspecified free variable and is therefore an open sentence. It requires assignment, substitution, or quantification before evaluation.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “Is x greater than 0?” a proposition? Classify it only; do not decide its truth or proof status.
No
It is an interrogative sentence rather than a closed statement that is directly asserted or denied.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “Does 0 belong to N?” a proposition? Classify it only; do not decide its truth or proof status.
No
It is an interrogative sentence rather than a closed statement that is directly asserted or denied.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “How many elements does this set have?” a proposition? Classify it only; do not decide its truth or proof status.
No
It is an interrogative sentence rather than a closed statement that is directly asserted or denied.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “Is f a function?” a proposition? Classify it only; do not decide its truth or proof status.
No
It is an interrogative sentence rather than a closed statement that is directly asserted or denied.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “Can you classify P?” a proposition? Classify it only; do not decide its truth or proof status.
No
It is an interrogative sentence rather than a closed statement that is directly asserted or denied.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “Set x to 0.” a proposition? Classify it only; do not decide its truth or proof status.
No
It is an imperative sentence rather than a closed statement that is directly asserted or denied.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “Calculate this distance.” a proposition? Classify it only; do not decide its truth or proof status.
No
It is an imperative sentence rather than a closed statement that is directly asserted or denied.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “Find the elements of set A.” a proposition? Classify it only; do not decide its truth or proof status.
No
It is an imperative sentence rather than a closed statement that is directly asserted or denied.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “Prove P.” a proposition? Classify it only; do not decide its truth or proof status.
No
It is an imperative sentence rather than a closed statement that is directly asserted or denied.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “Map the input into Y.” a proposition? Classify it only; do not decide its truth or proof status.
No
It is an imperative sentence rather than a closed statement that is directly asserted or denied.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “Every number equals itself.” a proposition? Classify it only; do not decide its truth or proof status.
Yes
It is a closed statement whose truth value can be discussed in the specified classical context. Being a proposition does not mean it has been proved true.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “z+1=4.” a proposition? Classify it only; do not decide its truth or proof status.
No
It contains an unspecified free variable and is therefore an open sentence. It requires assignment, substitution, or quantification before evaluation.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “Has P been proved?” a proposition? Classify it only; do not decide its truth or proof status.
No
It is an interrogative sentence rather than a closed statement that is directly asserted or denied.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “Do not treat P as proved.” a proposition? Classify it only; do not decide its truth or proof status.
No
It is an imperative sentence rather than a closed statement that is directly asserted or denied.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
In the introductory classical-logic context specified here, is “There exists a number equal to 0.” a proposition? Classify it only; do not decide its truth or proof status.
Yes
It is a closed statement whose truth value can be discussed in the specified classical context. Being a proposition does not mean it has been proved true.
Reviewed step-by-step derivation
Proposition · qf-proposition-classification · candidate
Let X={x1-1, x1-2, x1-3, x1-4} and Y={y1-1, y1-2, y1-3, y1-4, y1-5}. The finite correspondence is x1-1↦y1-1, x1-2↦y1-1, x1-3↦y1-1, x1-4↦y1-1. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
Yes
Every input in X has exactly one output, and every output belongs to Y. The actual range may be a proper subset of Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x2-1, x2-2, x2-3, x2-4} and Y={y2-1, y2-2, y2-3, y2-4, y2-5}. The finite correspondence is x2-1↦y2-1, x2-2↦y2-1, x2-3↦y2-1, x2-4↦y2-1, x2-1↦y2-2. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one input is assigned more than one output, so the correspondence is not a function.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x3-1, x3-2, x3-3, x3-4} and Y={y3-1, y3-2, y3-3, y3-4, y3-5}. The finite correspondence is x3-2↦y3-1, x3-3↦y3-1, x3-4↦y3-1. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one input in X has no assigned output, so the correspondence is not a total function X→Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x4-1, x4-2, x4-3, x4-4} and Y={y4-1, y4-2, y4-3, y4-4, y4-5}. The finite correspondence is x4-1↦z4, x4-2↦y4-1, x4-3↦y4-1, x4-4↦y4-1. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one assigned output lies outside the declared codomain Y, so the correspondence does not define a function X→Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x5-1, x5-2, x5-3, x5-4} and Y={y5-1, y5-2, y5-3, y5-4, y5-5}. The finite correspondence is x5-1↦y5-1, x5-2↦y5-1, x5-3↦y5-1, x5-4↦y5-2. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
Yes
Every input in X has exactly one output, and every output belongs to Y. The actual range may be a proper subset of Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x6-1, x6-2, x6-3, x6-4} and Y={y6-1, y6-2, y6-3, y6-4, y6-5}. The finite correspondence is x6-1↦y6-1, x6-2↦y6-1, x6-3↦y6-1, x6-4↦y6-2, x6-2↦y6-2. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one input is assigned more than one output, so the correspondence is not a function.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x7-1, x7-2, x7-3, x7-4} and Y={y7-1, y7-2, y7-3, y7-4, y7-5}. The finite correspondence is x7-1↦y7-1, x7-3↦y7-1, x7-4↦y7-2. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one input in X has no assigned output, so the correspondence is not a total function X→Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x8-1, x8-2, x8-3, x8-4} and Y={y8-1, y8-2, y8-3, y8-4, y8-5}. The finite correspondence is x8-1↦y8-1, x8-2↦z8, x8-3↦y8-1, x8-4↦y8-2. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one assigned output lies outside the declared codomain Y, so the correspondence does not define a function X→Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x9-1, x9-2, x9-3, x9-4} and Y={y9-1, y9-2, y9-3, y9-4, y9-5}. The finite correspondence is x9-1↦y9-1, x9-2↦y9-1, x9-3↦y9-2, x9-4↦y9-1. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
Yes
Every input in X has exactly one output, and every output belongs to Y. The actual range may be a proper subset of Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x10-1, x10-2, x10-3, x10-4} and Y={y10-1, y10-2, y10-3, y10-4, y10-5}. The finite correspondence is x10-1↦y10-1, x10-2↦y10-1, x10-3↦y10-2, x10-4↦y10-1, x10-3↦y10-3. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one input is assigned more than one output, so the correspondence is not a function.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x11-1, x11-2, x11-3, x11-4} and Y={y11-1, y11-2, y11-3, y11-4, y11-5}. The finite correspondence is x11-1↦y11-1, x11-2↦y11-1, x11-4↦y11-1. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one input in X has no assigned output, so the correspondence is not a total function X→Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x12-1, x12-2, x12-3, x12-4} and Y={y12-1, y12-2, y12-3, y12-4, y12-5}. The finite correspondence is x12-1↦y12-1, x12-2↦y12-1, x12-3↦z12, x12-4↦y12-1. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one assigned output lies outside the declared codomain Y, so the correspondence does not define a function X→Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x13-1, x13-2, x13-3, x13-4} and Y={y13-1, y13-2, y13-3, y13-4, y13-5}. The finite correspondence is x13-1↦y13-1, x13-2↦y13-1, x13-3↦y13-2, x13-4↦y13-2. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
Yes
Every input in X has exactly one output, and every output belongs to Y. The actual range may be a proper subset of Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x14-1, x14-2, x14-3, x14-4} and Y={y14-1, y14-2, y14-3, y14-4, y14-5}. The finite correspondence is x14-1↦y14-1, x14-2↦y14-1, x14-3↦y14-2, x14-4↦y14-2, x14-4↦y14-3. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one input is assigned more than one output, so the correspondence is not a function.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x15-1, x15-2, x15-3, x15-4} and Y={y15-1, y15-2, y15-3, y15-4, y15-5}. The finite correspondence is x15-1↦y15-1, x15-2↦y15-1, x15-3↦y15-2. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one input in X has no assigned output, so the correspondence is not a total function X→Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x16-1, x16-2, x16-3, x16-4} and Y={y16-1, y16-2, y16-3, y16-4, y16-5}. The finite correspondence is x16-1↦y16-1, x16-2↦y16-1, x16-3↦y16-2, x16-4↦z16. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one assigned output lies outside the declared codomain Y, so the correspondence does not define a function X→Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x17-1, x17-2, x17-3, x17-4} and Y={y17-1, y17-2, y17-3, y17-4, y17-5}. The finite correspondence is x17-1↦y17-1, x17-2↦y17-1, x17-3↦y17-2, x17-4↦y17-3. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
Yes
Every input in X has exactly one output, and every output belongs to Y. The actual range may be a proper subset of Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x18-1, x18-2, x18-3, x18-4} and Y={y18-1, y18-2, y18-3, y18-4, y18-5}. The finite correspondence is x18-1↦y18-1, x18-2↦y18-1, x18-3↦y18-2, x18-4↦y18-3, x18-1↦y18-2. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one input is assigned more than one output, so the correspondence is not a function.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x19-1, x19-2, x19-3, x19-4} and Y={y19-1, y19-2, y19-3, y19-4, y19-5}. The finite correspondence is x19-2↦y19-1, x19-3↦y19-2, x19-4↦y19-3. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one input in X has no assigned output, so the correspondence is not a total function X→Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x20-1, x20-2, x20-3, x20-4} and Y={y20-1, y20-2, y20-3, y20-4, y20-5}. The finite correspondence is x20-1↦z20, x20-2↦y20-1, x20-3↦y20-2, x20-4↦y20-3. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one assigned output lies outside the declared codomain Y, so the correspondence does not define a function X→Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x21-1, x21-2, x21-3, x21-4} and Y={y21-1, y21-2, y21-3, y21-4, y21-5}. The finite correspondence is x21-1↦y21-1, x21-2↦y21-2, x21-3↦y21-1, x21-4↦y21-1. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
Yes
Every input in X has exactly one output, and every output belongs to Y. The actual range may be a proper subset of Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x22-1, x22-2, x22-3, x22-4} and Y={y22-1, y22-2, y22-3, y22-4, y22-5}. The finite correspondence is x22-1↦y22-1, x22-2↦y22-2, x22-3↦y22-1, x22-4↦y22-1, x22-2↦y22-3. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one input is assigned more than one output, so the correspondence is not a function.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x23-1, x23-2, x23-3, x23-4} and Y={y23-1, y23-2, y23-3, y23-4, y23-5}. The finite correspondence is x23-1↦y23-1, x23-3↦y23-1, x23-4↦y23-1. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one input in X has no assigned output, so the correspondence is not a total function X→Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x24-1, x24-2, x24-3, x24-4} and Y={y24-1, y24-2, y24-3, y24-4, y24-5}. The finite correspondence is x24-1↦y24-1, x24-2↦z24, x24-3↦y24-1, x24-4↦y24-1. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one assigned output lies outside the declared codomain Y, so the correspondence does not define a function X→Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x25-1, x25-2, x25-3, x25-4} and Y={y25-1, y25-2, y25-3, y25-4, y25-5}. The finite correspondence is x25-1↦y25-1, x25-2↦y25-2, x25-3↦y25-1, x25-4↦y25-2. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
Yes
Every input in X has exactly one output, and every output belongs to Y. The actual range may be a proper subset of Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x26-1, x26-2, x26-3, x26-4} and Y={y26-1, y26-2, y26-3, y26-4, y26-5}. The finite correspondence is x26-1↦y26-1, x26-2↦y26-2, x26-3↦y26-1, x26-4↦y26-2, x26-3↦y26-2. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one input is assigned more than one output, so the correspondence is not a function.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x27-1, x27-2, x27-3, x27-4} and Y={y27-1, y27-2, y27-3, y27-4, y27-5}. The finite correspondence is x27-1↦y27-1, x27-2↦y27-2, x27-4↦y27-2. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one input in X has no assigned output, so the correspondence is not a total function X→Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x28-1, x28-2, x28-3, x28-4} and Y={y28-1, y28-2, y28-3, y28-4, y28-5}. The finite correspondence is x28-1↦y28-1, x28-2↦y28-2, x28-3↦z28, x28-4↦y28-2. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one assigned output lies outside the declared codomain Y, so the correspondence does not define a function X→Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x29-1, x29-2, x29-3, x29-4} and Y={y29-1, y29-2, y29-3, y29-4, y29-5}. The finite correspondence is x29-1↦y29-1, x29-2↦y29-2, x29-3↦y29-1, x29-4↦y29-3. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
Yes
Every input in X has exactly one output, and every output belongs to Y. The actual range may be a proper subset of Y.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Let X={x30-1, x30-2, x30-3, x30-4} and Y={y30-1, y30-2, y30-3, y30-4, y30-5}. The finite correspondence is x30-1↦y30-1, x30-2↦y30-2, x30-3↦y30-1, x30-4↦y30-3, x30-4↦y30-4. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.
No
At least one input is assigned more than one output, so the correspondence is not a function.
Reviewed step-by-step derivation
Function mapping · qf-finite-function-mapping · candidate
Canonical question JSONL · English JSONL · Validation report