# Foundational Mathematical Language

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.



## 1. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does 0 belong to N?

View answer and explanationYes

0 is a nonnegative integer, so it belongs to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0001/)

Natural number · qf-natural-number-membership · candidate





## 2. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does 1 belong to N?

View answer and explanationYes

1 is a nonnegative integer, so it belongs to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0002/)

Natural number · qf-natural-number-membership · candidate





## 3. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does 2 belong to N?

View answer and explanationYes

2 is a nonnegative integer, so it belongs to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0003/)

Natural number · qf-natural-number-membership · candidate





## 4. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does 5 belong to N?

View answer and explanationYes

5 is a nonnegative integer, so it belongs to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0004/)

Natural number · qf-natural-number-membership · candidate





## 5. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does 10 belong to N?

View answer and explanationYes

10 is a nonnegative integer, so it belongs to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0005/)

Natural number · qf-natural-number-membership · candidate





## 6. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does 25 belong to N?

View answer and explanationYes

25 is a nonnegative integer, so it belongs to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0006/)

Natural number · qf-natural-number-membership · candidate





## 7. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does 100 belong to N?

View answer and explanationYes

100 is a nonnegative integer, so it belongs to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0007/)

Natural number · qf-natural-number-membership · candidate





## 8. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does 7 belong to N?

View answer and explanationYes

7 is a nonnegative integer, so it belongs to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0008/)

Natural number · qf-natural-number-membership · candidate





## 9. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does 3 belong to N?

View answer and explanationYes

3 is a nonnegative integer, so it belongs to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0009/)

Natural number · qf-natural-number-membership · candidate





## 10. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does 12 belong to N?

View answer and explanationYes

12 is a nonnegative integer, so it belongs to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0010/)

Natural number · qf-natural-number-membership · candidate





## 11. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does -1 belong to N?

View answer and explanationNo

-1 is not a nonnegative integer, so it does not belong to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0011/)

Natural number · qf-natural-number-membership · candidate





## 12. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does -2 belong to N?

View answer and explanationNo

-2 is not a nonnegative integer, so it does not belong to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0012/)

Natural number · qf-natural-number-membership · candidate





## 13. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does -10 belong to N?

View answer and explanationNo

-10 is not a nonnegative integer, so it does not belong to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0013/)

Natural number · qf-natural-number-membership · candidate





## 14. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does -25 belong to N?

View answer and explanationNo

-25 is not a nonnegative integer, so it does not belong to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0014/)

Natural number · qf-natural-number-membership · candidate





## 15. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does -100 belong to N?

View answer and explanationNo

-100 is not a nonnegative integer, so it does not belong to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0015/)

Natural number · qf-natural-number-membership · candidate





## 16. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does -0.5 belong to N?

View answer and explanationNo

-0.5 is not a nonnegative integer, so it does not belong to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0016/)

Natural number · qf-natural-number-membership · candidate





## 17. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does 0.5 belong to N?

View answer and explanationNo

0.5 is not a nonnegative integer, so it does not belong to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0017/)

Natural number · qf-natural-number-membership · candidate





## 18. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does 1.5 belong to N?

View answer and explanationNo

1.5 is not a nonnegative integer, so it does not belong to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0018/)

Natural number · qf-natural-number-membership · candidate





## 19. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does 2.25 belong to N?

View answer and explanationNo

2.25 is not a nonnegative integer, so it does not belong to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0019/)

Natural number · qf-natural-number-membership · candidate





## 20. Natural-Number Membership

Under this OCME entry's convention that N includes 0, does -3.75 belong to N?

View answer and explanationNo

-3.75 is not a nonnegative integer, so it does not belong to N under this entry's convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-natural-number-membership-0020/)

Natural number · qf-natural-number-membership · candidate





## 21. Finite Set Membership

Let A={1, 2} be an explicitly enumerated finite set. Does 2 satisfy 2∈A?

View answer and explanationYes

2 occurs in the explicit enumeration of A, so 2∈A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0001/)

Set membership · qf-finite-set-membership · candidate





## 22. Finite Set Membership

Let A={11, 12, 13} be an explicitly enumerated finite set. Does 18 satisfy 18∈A?

View answer and explanationNo

18 does not occur in the explicit enumeration of A, so 18∉A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0002/)

Set membership · qf-finite-set-membership · candidate





## 23. Finite Set Membership

Let A={21, 22, 23, 24} be an explicitly enumerated finite set. Does 24 satisfy 24∈A?

View answer and explanationYes

24 occurs in the explicit enumeration of A, so 24∈A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0003/)

Set membership · qf-finite-set-membership · candidate





## 24. Finite Set Membership

Let A={31, 32} be an explicitly enumerated finite set. Does 37 satisfy 37∈A?

View answer and explanationNo

37 does not occur in the explicit enumeration of A, so 37∉A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0004/)

Set membership · qf-finite-set-membership · candidate





## 25. Finite Set Membership

Let A={41, 42, 43} be an explicitly enumerated finite set. Does 43 satisfy 43∈A?

View answer and explanationYes

43 occurs in the explicit enumeration of A, so 43∈A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0005/)

Set membership · qf-finite-set-membership · candidate





## 26. Finite Set Membership

Let A={51, 52, 53, 54} be an explicitly enumerated finite set. Does 59 satisfy 59∈A?

View answer and explanationNo

59 does not occur in the explicit enumeration of A, so 59∉A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0006/)

Set membership · qf-finite-set-membership · candidate





## 27. Finite Set Membership

Let A={61, 62} be an explicitly enumerated finite set. Does 62 satisfy 62∈A?

View answer and explanationYes

62 occurs in the explicit enumeration of A, so 62∈A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0007/)

Set membership · qf-finite-set-membership · candidate





## 28. Finite Set Membership

Let A={71, 72, 73} be an explicitly enumerated finite set. Does 78 satisfy 78∈A?

View answer and explanationNo

78 does not occur in the explicit enumeration of A, so 78∉A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0008/)

Set membership · qf-finite-set-membership · candidate





## 29. Finite Set Membership

Let A={81, 82, 83, 84} be an explicitly enumerated finite set. Does 82 satisfy 82∈A?

View answer and explanationYes

82 occurs in the explicit enumeration of A, so 82∈A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0009/)

Set membership · qf-finite-set-membership · candidate





## 30. Finite Set Membership

Let A={91, 92} be an explicitly enumerated finite set. Does 97 satisfy 97∈A?

View answer and explanationNo

97 does not occur in the explicit enumeration of A, so 97∉A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0010/)

Set membership · qf-finite-set-membership · candidate





## 31. Finite Set Membership

Let A={101, 102, 103} be an explicitly enumerated finite set. Does 103 satisfy 103∈A?

View answer and explanationYes

103 occurs in the explicit enumeration of A, so 103∈A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0011/)

Set membership · qf-finite-set-membership · candidate





## 32. Finite Set Membership

Let A={111, 112, 113, 114} be an explicitly enumerated finite set. Does 119 satisfy 119∈A?

View answer and explanationNo

119 does not occur in the explicit enumeration of A, so 119∉A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0012/)

Set membership · qf-finite-set-membership · candidate





## 33. Finite Set Membership

Let A={121, 122} be an explicitly enumerated finite set. Does 122 satisfy 122∈A?

View answer and explanationYes

122 occurs in the explicit enumeration of A, so 122∈A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0013/)

Set membership · qf-finite-set-membership · candidate





## 34. Finite Set Membership

Let A={131, 132, 133} be an explicitly enumerated finite set. Does 138 satisfy 138∈A?

View answer and explanationNo

138 does not occur in the explicit enumeration of A, so 138∉A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0014/)

Set membership · qf-finite-set-membership · candidate





## 35. Finite Set Membership

Let A={141, 142, 143, 144} be an explicitly enumerated finite set. Does 144 satisfy 144∈A?

View answer and explanationYes

144 occurs in the explicit enumeration of A, so 144∈A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0015/)

Set membership · qf-finite-set-membership · candidate





## 36. Finite Set Membership

Let A={151, 152} be an explicitly enumerated finite set. Does 157 satisfy 157∈A?

View answer and explanationNo

157 does not occur in the explicit enumeration of A, so 157∉A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0016/)

Set membership · qf-finite-set-membership · candidate





## 37. Finite Set Membership

Let A={161, 162, 163} be an explicitly enumerated finite set. Does 163 satisfy 163∈A?

View answer and explanationYes

163 occurs in the explicit enumeration of A, so 163∈A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0017/)

Set membership · qf-finite-set-membership · candidate





## 38. Finite Set Membership

Let A={171, 172, 173, 174} be an explicitly enumerated finite set. Does 179 satisfy 179∈A?

View answer and explanationNo

179 does not occur in the explicit enumeration of A, so 179∉A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0018/)

Set membership · qf-finite-set-membership · candidate





## 39. Finite Set Membership

Let A={181, 182} be an explicitly enumerated finite set. Does 182 satisfy 182∈A?

View answer and explanationYes

182 occurs in the explicit enumeration of A, so 182∈A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0019/)

Set membership · qf-finite-set-membership · candidate





## 40. Finite Set Membership

Let A={191, 192, 193} be an explicitly enumerated finite set. Does 198 satisfy 198∈A?

View answer and explanationNo

198 does not occur in the explicit enumeration of A, so 198∉A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0020/)

Set membership · qf-finite-set-membership · candidate





## 41. Finite Set Membership

Let A={201, 202, 203, 204} be an explicitly enumerated finite set. Does 202 satisfy 202∈A?

View answer and explanationYes

202 occurs in the explicit enumeration of A, so 202∈A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0021/)

Set membership · qf-finite-set-membership · candidate





## 42. Finite Set Membership

Let A={211, 212} be an explicitly enumerated finite set. Does 217 satisfy 217∈A?

View answer and explanationNo

217 does not occur in the explicit enumeration of A, so 217∉A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0022/)

Set membership · qf-finite-set-membership · candidate





## 43. Finite Set Membership

Let A={221, 222, 223} be an explicitly enumerated finite set. Does 223 satisfy 223∈A?

View answer and explanationYes

223 occurs in the explicit enumeration of A, so 223∈A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0023/)

Set membership · qf-finite-set-membership · candidate





## 44. Finite Set Membership

Let A={231, 232, 233, 234} be an explicitly enumerated finite set. Does 239 satisfy 239∈A?

View answer and explanationNo

239 does not occur in the explicit enumeration of A, so 239∉A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0024/)

Set membership · qf-finite-set-membership · candidate





## 45. Finite Set Membership

Let A={241, 242} be an explicitly enumerated finite set. Does 242 satisfy 242∈A?

View answer and explanationYes

242 occurs in the explicit enumeration of A, so 242∈A.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-set-membership-0025/)

Set membership · qf-finite-set-membership · candidate





## 46. Proposition Classification

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0001/)

Proposition · qf-proposition-classification · candidate





## 47. Proposition Classification

In the introductory classical-logic context specified here, is “7=7.” a proposition? Classify it only; do not decide its truth or proof status.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0002/)

Proposition · qf-proposition-classification · candidate





## 48. Proposition Classification

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0003/)

Proposition · qf-proposition-classification · candidate





## 49. Proposition Classification

In the introductory classical-logic context specified here, is “0=1.” a proposition? Classify it only; do not decide its truth or proof status.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0004/)

Proposition · qf-proposition-classification · candidate





## 50. Proposition Classification

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0005/)

Proposition · qf-proposition-classification · candidate





## 51. Proposition Classification

In the introductory classical-logic context specified here, is “x>0.” a proposition? Classify it only; do not decide its truth or proof status.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0006/)

Proposition · qf-proposition-classification · candidate





## 52. Proposition Classification

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0007/)

Proposition · qf-proposition-classification · candidate





## 53. Proposition Classification

In the introductory classical-logic context specified here, is “y∈A.” a proposition? Classify it only; do not decide its truth or proof status.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0008/)

Proposition · qf-proposition-classification · candidate





## 54. Proposition Classification

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0009/)

Proposition · qf-proposition-classification · candidate





## 55. Proposition Classification

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0010/)

Proposition · qf-proposition-classification · candidate





## 56. Proposition Classification

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.

View answer and explanationNo

It is an interrogative sentence rather than a closed statement that is directly asserted or denied.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0011/)

Proposition · qf-proposition-classification · candidate





## 57. Proposition Classification

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.

View answer and explanationNo

It is an interrogative sentence rather than a closed statement that is directly asserted or denied.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0012/)

Proposition · qf-proposition-classification · candidate





## 58. Proposition Classification

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.

View answer and explanationNo

It is an interrogative sentence rather than a closed statement that is directly asserted or denied.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0013/)

Proposition · qf-proposition-classification · candidate





## 59. Proposition Classification

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.

View answer and explanationNo

It is an interrogative sentence rather than a closed statement that is directly asserted or denied.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0014/)

Proposition · qf-proposition-classification · candidate





## 60. Proposition Classification

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.

View answer and explanationNo

It is an interrogative sentence rather than a closed statement that is directly asserted or denied.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0015/)

Proposition · qf-proposition-classification · candidate





## 61. Proposition Classification

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.

View answer and explanationNo

It is an imperative sentence rather than a closed statement that is directly asserted or denied.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0016/)

Proposition · qf-proposition-classification · candidate





## 62. Proposition Classification

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.

View answer and explanationNo

It is an imperative sentence rather than a closed statement that is directly asserted or denied.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0017/)

Proposition · qf-proposition-classification · candidate





## 63. Proposition Classification

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.

View answer and explanationNo

It is an imperative sentence rather than a closed statement that is directly asserted or denied.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0018/)

Proposition · qf-proposition-classification · candidate





## 64. Proposition Classification

In the introductory classical-logic context specified here, is “Prove P.” a proposition? Classify it only; do not decide its truth or proof status.

View answer and explanationNo

It is an imperative sentence rather than a closed statement that is directly asserted or denied.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0019/)

Proposition · qf-proposition-classification · candidate





## 65. Proposition Classification

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.

View answer and explanationNo

It is an imperative sentence rather than a closed statement that is directly asserted or denied.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0020/)

Proposition · qf-proposition-classification · candidate





## 66. Proposition Classification

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0021/)

Proposition · qf-proposition-classification · candidate





## 67. Proposition Classification

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0022/)

Proposition · qf-proposition-classification · candidate





## 68. Proposition Classification

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.

View answer and explanationNo

It is an interrogative sentence rather than a closed statement that is directly asserted or denied.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0023/)

Proposition · qf-proposition-classification · candidate





## 69. Proposition Classification

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.

View answer and explanationNo

It is an imperative sentence rather than a closed statement that is directly asserted or denied.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0024/)

Proposition · qf-proposition-classification · candidate





## 70. Proposition Classification

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20260913-proposition-classification-0025/)

Proposition · qf-proposition-classification · candidate





## 71. Finite Mapping as a Function

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0001/)

Function mapping · qf-finite-function-mapping · candidate





## 72. Finite Mapping as a Function

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.

View answer and explanationNo

At least one input is assigned more than one output, so the correspondence is not a function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0002/)

Function mapping · qf-finite-function-mapping · candidate





## 73. Finite Mapping as a Function

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0003/)

Function mapping · qf-finite-function-mapping · candidate





## 74. Finite Mapping as a Function

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0004/)

Function mapping · qf-finite-function-mapping · candidate





## 75. Finite Mapping as a Function

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0005/)

Function mapping · qf-finite-function-mapping · candidate





## 76. Finite Mapping as a Function

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.

View answer and explanationNo

At least one input is assigned more than one output, so the correspondence is not a function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0006/)

Function mapping · qf-finite-function-mapping · candidate





## 77. Finite Mapping as a Function

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0007/)

Function mapping · qf-finite-function-mapping · candidate





## 78. Finite Mapping as a Function

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0008/)

Function mapping · qf-finite-function-mapping · candidate





## 79. Finite Mapping as a Function

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0009/)

Function mapping · qf-finite-function-mapping · candidate





## 80. Finite Mapping as a Function

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.

View answer and explanationNo

At least one input is assigned more than one output, so the correspondence is not a function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0010/)

Function mapping · qf-finite-function-mapping · candidate





## 81. Finite Mapping as a Function

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0011/)

Function mapping · qf-finite-function-mapping · candidate





## 82. Finite Mapping as a Function

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0012/)

Function mapping · qf-finite-function-mapping · candidate





## 83. Finite Mapping as a Function

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0013/)

Function mapping · qf-finite-function-mapping · candidate





## 84. Finite Mapping as a Function

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.

View answer and explanationNo

At least one input is assigned more than one output, so the correspondence is not a function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0014/)

Function mapping · qf-finite-function-mapping · candidate





## 85. Finite Mapping as a Function

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0015/)

Function mapping · qf-finite-function-mapping · candidate





## 86. Finite Mapping as a Function

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0016/)

Function mapping · qf-finite-function-mapping · candidate





## 87. Finite Mapping as a Function

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0017/)

Function mapping · qf-finite-function-mapping · candidate





## 88. Finite Mapping as a Function

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.

View answer and explanationNo

At least one input is assigned more than one output, so the correspondence is not a function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0018/)

Function mapping · qf-finite-function-mapping · candidate





## 89. Finite Mapping as a Function

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0019/)

Function mapping · qf-finite-function-mapping · candidate





## 90. Finite Mapping as a Function

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0020/)

Function mapping · qf-finite-function-mapping · candidate





## 91. Finite Mapping as a Function

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0021/)

Function mapping · qf-finite-function-mapping · candidate





## 92. Finite Mapping as a Function

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.

View answer and explanationNo

At least one input is assigned more than one output, so the correspondence is not a function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0022/)

Function mapping · qf-finite-function-mapping · candidate





## 93. Finite Mapping as a Function

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0023/)

Function mapping · qf-finite-function-mapping · candidate





## 94. Finite Mapping as a Function

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0024/)

Function mapping · qf-finite-function-mapping · candidate





## 95. Finite Mapping as a Function

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0025/)

Function mapping · qf-finite-function-mapping · candidate





## 96. Finite Mapping as a Function

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.

View answer and explanationNo

At least one input is assigned more than one output, so the correspondence is not a function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0026/)

Function mapping · qf-finite-function-mapping · candidate





## 97. Finite Mapping as a Function

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0027/)

Function mapping · qf-finite-function-mapping · candidate





## 98. Finite Mapping as a Function

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0028/)

Function mapping · qf-finite-function-mapping · candidate





## 99. Finite Mapping as a Function

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0029/)

Function mapping · qf-finite-function-mapping · candidate





## 100. Finite Mapping as a Function

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.

View answer and explanationNo

At least one input is assigned more than one output, so the correspondence is not a function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260913-finite-function-mapping-0030/)

Function mapping · qf-finite-function-mapping · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20260913-foundational-language/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20260913-foundational-language.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20260913-foundational-language/validation-report.json)