# Mapping-Bound Injection, Surjection, Bijection and Finite Pigeonhole Certificates

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-20261014-advance-20261030-r05-cardinality-comparison-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[0, 1] and target=[100, 101, 102] as distinct-element lists in their original orders, with mapping=[100, 101] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([0, 1], [100, 101, 102], [100, 101], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([0, 1], [100, 101, 102], [100, 101], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0001/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 2. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[10, 11, 12] and target=[110, 111, 112, 113] as distinct-element lists in their original orders, with mapping=[110, 111, 112] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([10, 11, 12], [110, 111, 112, 113], [110, 111, 112], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([10, 11, 12], [110, 111, 112, 113], [110, 111, 112], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0002/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 3. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[20, 21, 22, 23] and target=[120, 121, 122, 123, 124] as distinct-element lists in their original orders, with mapping=[120, 121, 122, 123] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([20, 21, 22, 23], [120, 121, 122, 123, 124], [120, 121, 122, 123], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([20, 21, 22, 23], [120, 121, 122, 123, 124], [120, 121, 122, 123], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0003/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 4. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[30, 31, 32, 33, 34] and target=[130, 131] as distinct-element lists in their original orders, with mapping=[130, 131, 130, 131, 130] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([30, 31, 32, 33, 34], [130, 131], [130, 131, 130, 131, 130], No, no_comparison_certificate)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([30, 31, 32, 33, 34], [130, 131], [130, 131, 130, 131, 130], No, no_comparison_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0004/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 5. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[40, 41] and target=[140, 141, 142] as distinct-element lists in their original orders, with mapping=[140, 141] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([40, 41], [140, 141, 142], [140, 141], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([40, 41], [140, 141, 142], [140, 141], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0005/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 6. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[50, 51, 52] and target=[150, 151, 152, 153] as distinct-element lists in their original orders, with mapping=[150, 151, 152] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([50, 51, 52], [150, 151, 152, 153], [150, 151, 152], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([50, 51, 52], [150, 151, 152, 153], [150, 151, 152], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0006/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 7. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[60, 61, 62, 63] and target=[160, 161, 162, 163, 164] as distinct-element lists in their original orders, with mapping=[160, 161, 162, 163] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([60, 61, 62, 63], [160, 161, 162, 163, 164], [160, 161, 162, 163], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([60, 61, 62, 63], [160, 161, 162, 163, 164], [160, 161, 162, 163], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0007/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 8. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[70, 71, 72, 73, 74] and target=[170, 171] as distinct-element lists in their original orders, with mapping=[170, 171, 170, 171, 170] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([70, 71, 72, 73, 74], [170, 171], [170, 171, 170, 171, 170], No, no_comparison_certificate)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([70, 71, 72, 73, 74], [170, 171], [170, 171, 170, 171, 170], No, no_comparison_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0008/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 9. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[80, 81] and target=[180, 181, 182] as distinct-element lists in their original orders, with mapping=[180, 181] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([80, 81], [180, 181, 182], [180, 181], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([80, 81], [180, 181, 182], [180, 181], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0009/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 10. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[90, 91, 92] and target=[190, 191, 192, 193] as distinct-element lists in their original orders, with mapping=[190, 191, 192] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([90, 91, 92], [190, 191, 192, 193], [190, 191, 192], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([90, 91, 92], [190, 191, 192, 193], [190, 191, 192], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0010/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 11. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[100, 101, 102, 103] and target=[200, 201, 202, 203, 204] as distinct-element lists in their original orders, with mapping=[200, 201, 202, 203] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([100, 101, 102, 103], [200, 201, 202, 203, 204], [200, 201, 202, 203], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([100, 101, 102, 103], [200, 201, 202, 203, 204], [200, 201, 202, 203], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0011/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 12. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[110, 111, 112, 113, 114] and target=[210, 211] as distinct-element lists in their original orders, with mapping=[210, 211, 210, 211, 210] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([110, 111, 112, 113, 114], [210, 211], [210, 211, 210, 211, 210], No, no_comparison_certificate)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([110, 111, 112, 113, 114], [210, 211], [210, 211, 210, 211, 210], No, no_comparison_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0012/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 13. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[120, 121] and target=[220, 221, 222] as distinct-element lists in their original orders, with mapping=[220, 221] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([120, 121], [220, 221, 222], [220, 221], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([120, 121], [220, 221, 222], [220, 221], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0013/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 14. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[130, 131, 132] and target=[230, 231, 232, 233] as distinct-element lists in their original orders, with mapping=[230, 231, 232] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([130, 131, 132], [230, 231, 232, 233], [230, 231, 232], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([130, 131, 132], [230, 231, 232, 233], [230, 231, 232], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0014/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 15. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[140, 141, 142, 143] and target=[240, 241, 242, 243, 244] as distinct-element lists in their original orders, with mapping=[240, 241, 242, 243] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([140, 141, 142, 143], [240, 241, 242, 243, 244], [240, 241, 242, 243], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([140, 141, 142, 143], [240, 241, 242, 243, 244], [240, 241, 242, 243], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0015/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 16. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[150, 151, 152, 153, 154] and target=[250, 251] as distinct-element lists in their original orders, with mapping=[250, 251, 250, 251, 250] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([150, 151, 152, 153, 154], [250, 251], [250, 251, 250, 251, 250], No, no_comparison_certificate)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([150, 151, 152, 153, 154], [250, 251], [250, 251, 250, 251, 250], No, no_comparison_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0016/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 17. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[160, 161] and target=[260, 261, 262] as distinct-element lists in their original orders, with mapping=[260, 261] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([160, 161], [260, 261, 262], [260, 261], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([160, 161], [260, 261, 262], [260, 261], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0017/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 18. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[170, 171, 172] and target=[270, 271, 272, 273] as distinct-element lists in their original orders, with mapping=[270, 271, 272] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([170, 171, 172], [270, 271, 272, 273], [270, 271, 272], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([170, 171, 172], [270, 271, 272, 273], [270, 271, 272], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0018/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 19. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[180, 181, 182, 183] and target=[280, 281, 282, 283, 284] as distinct-element lists in their original orders, with mapping=[280, 281, 282, 283] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([180, 181, 182, 183], [280, 281, 282, 283, 284], [280, 281, 282, 283], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([180, 181, 182, 183], [280, 281, 282, 283, 284], [280, 281, 282, 283], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0019/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 20. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[190, 191, 192, 193, 194] and target=[290, 291] as distinct-element lists in their original orders, with mapping=[290, 291, 290, 291, 290] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([190, 191, 192, 193, 194], [290, 291], [290, 291, 290, 291, 290], No, no_comparison_certificate)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([190, 191, 192, 193, 194], [290, 291], [290, 291, 290, 291, 290], No, no_comparison_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0020/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 21. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[200, 201] and target=[300, 301, 302] as distinct-element lists in their original orders, with mapping=[300, 301] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([200, 201], [300, 301, 302], [300, 301], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([200, 201], [300, 301, 302], [300, 301], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0021/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 22. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[210, 211, 212] and target=[310, 311, 312, 313] as distinct-element lists in their original orders, with mapping=[310, 311, 312] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([210, 211, 212], [310, 311, 312, 313], [310, 311, 312], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([210, 211, 212], [310, 311, 312, 313], [310, 311, 312], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0022/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 23. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[220, 221, 222, 223] and target=[320, 321, 322, 323, 324] as distinct-element lists in their original orders, with mapping=[320, 321, 322, 323] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([220, 221, 222, 223], [320, 321, 322, 323, 324], [320, 321, 322, 323], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([220, 221, 222, 223], [320, 321, 322, 323, 324], [320, 321, 322, 323], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0023/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 24. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[230, 231, 232, 233, 234] and target=[330, 331] as distinct-element lists in their original orders, with mapping=[330, 331, 330, 331, 330] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([230, 231, 232, 233, 234], [330, 331], [330, 331, 330, 331, 330], No, no_comparison_certificate)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([230, 231, 232, 233, 234], [330, 331], [330, 331, 330, 331, 330], No, no_comparison_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0024/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 25. An Injection Certifies One Finite Cardinality Inequality

Use the complete finite source=[240, 241] and target=[340, 341, 342] as distinct-element lists in their original orders, with mapping=[340, 341] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.

View answer and explanation([240, 241], [340, 341, 342], [340, 341], Yes, Yes)

Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([240, 241], [340, 341, 342], [340, 341], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0025/)

Cardinality comparison of sets · qf-mod30-cardinality-injection · candidate





## 26. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[0, 1] and target=[100, 101, 102] as distinct-element lists in their original orders, with mapping=[100, 101] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([0, 1], [100, 101, 102], [100, 101], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([0, 1], [100, 101, 102], [100, 101], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0026/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 27. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[10, 11, 12] and target=[110, 111, 112, 113] as distinct-element lists in their original orders, with mapping=[110, 111, 112] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([10, 11, 12], [110, 111, 112, 113], [110, 111, 112], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([10, 11, 12], [110, 111, 112, 113], [110, 111, 112], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0027/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 28. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[20, 21, 22, 23] and target=[120, 121, 122, 123, 124] as distinct-element lists in their original orders, with mapping=[120, 121, 122, 123] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([20, 21, 22, 23], [120, 121, 122, 123, 124], [120, 121, 122, 123], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([20, 21, 22, 23], [120, 121, 122, 123, 124], [120, 121, 122, 123], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0028/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 29. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[30, 31, 32, 33, 34] and target=[130, 131] as distinct-element lists in their original orders, with mapping=[130, 131, 130, 131, 130] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([30, 31, 32, 33, 34], [130, 131], [130, 131, 130, 131, 130], Yes, Yes, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([30, 31, 32, 33, 34], [130, 131], [130, 131, 130, 131, 130], Yes, Yes, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0029/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 30. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[40, 41] and target=[140, 141, 142] as distinct-element lists in their original orders, with mapping=[140, 141] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([40, 41], [140, 141, 142], [140, 141], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([40, 41], [140, 141, 142], [140, 141], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0030/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 31. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[50, 51, 52] and target=[150, 151, 152, 153] as distinct-element lists in their original orders, with mapping=[150, 151, 152] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([50, 51, 52], [150, 151, 152, 153], [150, 151, 152], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([50, 51, 52], [150, 151, 152, 153], [150, 151, 152], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0031/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 32. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[60, 61, 62, 63] and target=[160, 161, 162, 163, 164] as distinct-element lists in their original orders, with mapping=[160, 161, 162, 163] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([60, 61, 62, 63], [160, 161, 162, 163, 164], [160, 161, 162, 163], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([60, 61, 62, 63], [160, 161, 162, 163, 164], [160, 161, 162, 163], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0032/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 33. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[70, 71, 72, 73, 74] and target=[170, 171] as distinct-element lists in their original orders, with mapping=[170, 171, 170, 171, 170] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([70, 71, 72, 73, 74], [170, 171], [170, 171, 170, 171, 170], Yes, Yes, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([70, 71, 72, 73, 74], [170, 171], [170, 171, 170, 171, 170], Yes, Yes, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0033/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 34. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[80, 81] and target=[180, 181, 182] as distinct-element lists in their original orders, with mapping=[180, 181] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([80, 81], [180, 181, 182], [180, 181], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([80, 81], [180, 181, 182], [180, 181], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0034/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 35. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[90, 91, 92] and target=[190, 191, 192, 193] as distinct-element lists in their original orders, with mapping=[190, 191, 192] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([90, 91, 92], [190, 191, 192, 193], [190, 191, 192], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([90, 91, 92], [190, 191, 192, 193], [190, 191, 192], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0035/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 36. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[100, 101, 102, 103] and target=[200, 201, 202, 203, 204] as distinct-element lists in their original orders, with mapping=[200, 201, 202, 203] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([100, 101, 102, 103], [200, 201, 202, 203, 204], [200, 201, 202, 203], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([100, 101, 102, 103], [200, 201, 202, 203, 204], [200, 201, 202, 203], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0036/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 37. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[110, 111, 112, 113, 114] and target=[210, 211] as distinct-element lists in their original orders, with mapping=[210, 211, 210, 211, 210] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([110, 111, 112, 113, 114], [210, 211], [210, 211, 210, 211, 210], Yes, Yes, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([110, 111, 112, 113, 114], [210, 211], [210, 211, 210, 211, 210], Yes, Yes, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0037/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 38. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[120, 121] and target=[220, 221, 222] as distinct-element lists in their original orders, with mapping=[220, 221] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([120, 121], [220, 221, 222], [220, 221], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([120, 121], [220, 221, 222], [220, 221], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0038/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 39. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[130, 131, 132] and target=[230, 231, 232, 233] as distinct-element lists in their original orders, with mapping=[230, 231, 232] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([130, 131, 132], [230, 231, 232, 233], [230, 231, 232], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([130, 131, 132], [230, 231, 232, 233], [230, 231, 232], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0039/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 40. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[140, 141, 142, 143] and target=[240, 241, 242, 243, 244] as distinct-element lists in their original orders, with mapping=[240, 241, 242, 243] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([140, 141, 142, 143], [240, 241, 242, 243, 244], [240, 241, 242, 243], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([140, 141, 142, 143], [240, 241, 242, 243, 244], [240, 241, 242, 243], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0040/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 41. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[150, 151, 152, 153, 154] and target=[250, 251] as distinct-element lists in their original orders, with mapping=[250, 251, 250, 251, 250] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([150, 151, 152, 153, 154], [250, 251], [250, 251, 250, 251, 250], Yes, Yes, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([150, 151, 152, 153, 154], [250, 251], [250, 251, 250, 251, 250], Yes, Yes, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0041/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 42. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[160, 161] and target=[260, 261, 262] as distinct-element lists in their original orders, with mapping=[260, 261] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([160, 161], [260, 261, 262], [260, 261], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([160, 161], [260, 261, 262], [260, 261], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0042/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 43. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[170, 171, 172] and target=[270, 271, 272, 273] as distinct-element lists in their original orders, with mapping=[270, 271, 272] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([170, 171, 172], [270, 271, 272, 273], [270, 271, 272], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([170, 171, 172], [270, 271, 272, 273], [270, 271, 272], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0043/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 44. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[180, 181, 182, 183] and target=[280, 281, 282, 283, 284] as distinct-element lists in their original orders, with mapping=[280, 281, 282, 283] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([180, 181, 182, 183], [280, 281, 282, 283, 284], [280, 281, 282, 283], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([180, 181, 182, 183], [280, 281, 282, 283, 284], [280, 281, 282, 283], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0044/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 45. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[190, 191, 192, 193, 194] and target=[290, 291] as distinct-element lists in their original orders, with mapping=[290, 291, 290, 291, 290] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([190, 191, 192, 193, 194], [290, 291], [290, 291, 290, 291, 290], Yes, Yes, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([190, 191, 192, 193, 194], [290, 291], [290, 291, 290, 291, 290], Yes, Yes, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0045/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 46. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[200, 201] and target=[300, 301, 302] as distinct-element lists in their original orders, with mapping=[300, 301] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([200, 201], [300, 301, 302], [300, 301], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([200, 201], [300, 301, 302], [300, 301], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0046/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 47. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[210, 211, 212] and target=[310, 311, 312, 313] as distinct-element lists in their original orders, with mapping=[310, 311, 312] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([210, 211, 212], [310, 311, 312, 313], [310, 311, 312], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([210, 211, 212], [310, 311, 312, 313], [310, 311, 312], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0047/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 48. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[220, 221, 222, 223] and target=[320, 321, 322, 323, 324] as distinct-element lists in their original orders, with mapping=[320, 321, 322, 323] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([220, 221, 222, 223], [320, 321, 322, 323, 324], [320, 321, 322, 323], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([220, 221, 222, 223], [320, 321, 322, 323, 324], [320, 321, 322, 323], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0048/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 49. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[230, 231, 232, 233, 234] and target=[330, 331] as distinct-element lists in their original orders, with mapping=[330, 331, 330, 331, 330] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([230, 231, 232, 233, 234], [330, 331], [330, 331, 330, 331, 330], Yes, Yes, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([230, 231, 232, 233, 234], [330, 331], [330, 331, 330, 331, 330], Yes, Yes, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0049/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 50. A Surjection Supplies the Reverse Comparison Only in This Finite Scope

Use the complete finite source=[240, 241] and target=[340, 341, 342] as distinct-element lists in their original orders, with mapping=[340, 341] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.

View answer and explanation([240, 241], [340, 341, 342], [340, 341], No, no_comparison_certificate, finite_only)

Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([240, 241], [340, 341, 342], [340, 341], No, no_comparison_certificate, finite_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0050/)

Cardinality comparison of sets · qf-mod30-cardinality-surjection-finite · candidate





## 51. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[0, 1] and target=[100, 101, 102] as distinct-element lists in their original orders, with mapping=[100, 101] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([0, 1], [100, 101, 102], [100, 101], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([0, 1], [100, 101, 102], [100, 101], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0051/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 52. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[10, 11, 12] and target=[110, 111, 112, 113] as distinct-element lists in their original orders, with mapping=[110, 111, 112] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([10, 11, 12], [110, 111, 112, 113], [110, 111, 112], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([10, 11, 12], [110, 111, 112, 113], [110, 111, 112], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0052/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 53. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[20, 21, 22, 23] and target=[120, 121, 122, 123, 124] as distinct-element lists in their original orders, with mapping=[120, 121, 122, 123] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([20, 21, 22, 23], [120, 121, 122, 123, 124], [120, 121, 122, 123], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([20, 21, 22, 23], [120, 121, 122, 123, 124], [120, 121, 122, 123], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0053/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 54. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[30, 31, 32, 33, 34] and target=[130, 131] as distinct-element lists in their original orders, with mapping=[130, 131, 130, 131, 130] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([30, 31, 32, 33, 34], [130, 131], [130, 131, 130, 131, 130], No, Yes, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([30, 31, 32, 33, 34], [130, 131], [130, 131, 130, 131, 130], No, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0054/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 55. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[40, 41] and target=[140, 141, 142] as distinct-element lists in their original orders, with mapping=[140, 141] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([40, 41], [140, 141, 142], [140, 141], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([40, 41], [140, 141, 142], [140, 141], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0055/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 56. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[50, 51, 52] and target=[150, 151, 152, 153] as distinct-element lists in their original orders, with mapping=[150, 151, 152] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([50, 51, 52], [150, 151, 152, 153], [150, 151, 152], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([50, 51, 52], [150, 151, 152, 153], [150, 151, 152], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0056/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 57. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[60, 61, 62, 63] and target=[160, 161, 162, 163, 164] as distinct-element lists in their original orders, with mapping=[160, 161, 162, 163] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([60, 61, 62, 63], [160, 161, 162, 163, 164], [160, 161, 162, 163], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([60, 61, 62, 63], [160, 161, 162, 163, 164], [160, 161, 162, 163], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0057/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 58. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[70, 71, 72, 73, 74] and target=[170, 171] as distinct-element lists in their original orders, with mapping=[170, 171, 170, 171, 170] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([70, 71, 72, 73, 74], [170, 171], [170, 171, 170, 171, 170], No, Yes, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([70, 71, 72, 73, 74], [170, 171], [170, 171, 170, 171, 170], No, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0058/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 59. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[80, 81] and target=[180, 181, 182] as distinct-element lists in their original orders, with mapping=[180, 181] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([80, 81], [180, 181, 182], [180, 181], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([80, 81], [180, 181, 182], [180, 181], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0059/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 60. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[90, 91, 92] and target=[190, 191, 192, 193] as distinct-element lists in their original orders, with mapping=[190, 191, 192] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([90, 91, 92], [190, 191, 192, 193], [190, 191, 192], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([90, 91, 92], [190, 191, 192, 193], [190, 191, 192], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0060/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 61. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[100, 101, 102, 103] and target=[200, 201, 202, 203, 204] as distinct-element lists in their original orders, with mapping=[200, 201, 202, 203] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([100, 101, 102, 103], [200, 201, 202, 203, 204], [200, 201, 202, 203], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([100, 101, 102, 103], [200, 201, 202, 203, 204], [200, 201, 202, 203], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0061/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 62. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[110, 111, 112, 113, 114] and target=[210, 211] as distinct-element lists in their original orders, with mapping=[210, 211, 210, 211, 210] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([110, 111, 112, 113, 114], [210, 211], [210, 211, 210, 211, 210], No, Yes, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([110, 111, 112, 113, 114], [210, 211], [210, 211, 210, 211, 210], No, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0062/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 63. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[120, 121] and target=[220, 221, 222] as distinct-element lists in their original orders, with mapping=[220, 221] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([120, 121], [220, 221, 222], [220, 221], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([120, 121], [220, 221, 222], [220, 221], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0063/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 64. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[130, 131, 132] and target=[230, 231, 232, 233] as distinct-element lists in their original orders, with mapping=[230, 231, 232] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([130, 131, 132], [230, 231, 232, 233], [230, 231, 232], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([130, 131, 132], [230, 231, 232, 233], [230, 231, 232], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0064/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 65. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[140, 141, 142, 143] and target=[240, 241, 242, 243, 244] as distinct-element lists in their original orders, with mapping=[240, 241, 242, 243] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([140, 141, 142, 143], [240, 241, 242, 243, 244], [240, 241, 242, 243], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([140, 141, 142, 143], [240, 241, 242, 243, 244], [240, 241, 242, 243], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0065/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 66. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[150, 151, 152, 153, 154] and target=[250, 251] as distinct-element lists in their original orders, with mapping=[250, 251, 250, 251, 250] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([150, 151, 152, 153, 154], [250, 251], [250, 251, 250, 251, 250], No, Yes, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([150, 151, 152, 153, 154], [250, 251], [250, 251, 250, 251, 250], No, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0066/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 67. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[160, 161] and target=[260, 261, 262] as distinct-element lists in their original orders, with mapping=[260, 261] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([160, 161], [260, 261, 262], [260, 261], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([160, 161], [260, 261, 262], [260, 261], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0067/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 68. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[170, 171, 172] and target=[270, 271, 272, 273] as distinct-element lists in their original orders, with mapping=[270, 271, 272] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([170, 171, 172], [270, 271, 272, 273], [270, 271, 272], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([170, 171, 172], [270, 271, 272, 273], [270, 271, 272], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0068/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 69. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[180, 181, 182, 183] and target=[280, 281, 282, 283, 284] as distinct-element lists in their original orders, with mapping=[280, 281, 282, 283] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([180, 181, 182, 183], [280, 281, 282, 283, 284], [280, 281, 282, 283], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([180, 181, 182, 183], [280, 281, 282, 283, 284], [280, 281, 282, 283], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0069/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 70. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[190, 191, 192, 193, 194] and target=[290, 291] as distinct-element lists in their original orders, with mapping=[290, 291, 290, 291, 290] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([190, 191, 192, 193, 194], [290, 291], [290, 291, 290, 291, 290], No, Yes, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([190, 191, 192, 193, 194], [290, 291], [290, 291, 290, 291, 290], No, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0070/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 71. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[200, 201] and target=[300, 301, 302] as distinct-element lists in their original orders, with mapping=[300, 301] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([200, 201], [300, 301, 302], [300, 301], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([200, 201], [300, 301, 302], [300, 301], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0071/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 72. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[210, 211, 212] and target=[310, 311, 312, 313] as distinct-element lists in their original orders, with mapping=[310, 311, 312] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([210, 211, 212], [310, 311, 312, 313], [310, 311, 312], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([210, 211, 212], [310, 311, 312, 313], [310, 311, 312], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0072/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 73. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[220, 221, 222, 223] and target=[320, 321, 322, 323, 324] as distinct-element lists in their original orders, with mapping=[320, 321, 322, 323] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([220, 221, 222, 223], [320, 321, 322, 323, 324], [320, 321, 322, 323], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([220, 221, 222, 223], [320, 321, 322, 323, 324], [320, 321, 322, 323], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0073/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 74. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[230, 231, 232, 233, 234] and target=[330, 331] as distinct-element lists in their original orders, with mapping=[330, 331, 330, 331, 330] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([230, 231, 232, 233, 234], [330, 331], [330, 331, 330, 331, 330], No, Yes, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([230, 231, 232, 233, 234], [330, 331], [330, 331, 330, 331, 330], No, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0074/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 75. The Last Equality Flag Is a Bijection-Supported Certificate

Use the complete finite source=[240, 241] and target=[340, 341, 342] as distinct-element lists in their original orders, with mapping=[340, 341] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

View answer and explanation([240, 241], [340, 341, 342], [340, 341], Yes, No, No, No)

Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([240, 241], [340, 341, 342], [340, 341], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0075/)

Cardinality comparison of sets · qf-mod30-cardinality-bijection · candidate





## 76. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[0, 1] and target=[100, 101, 102] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(2, 3, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (2, 3, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0076/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 77. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[10, 11, 12] and target=[110, 111, 112, 113] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(3, 4, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (3, 4, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0077/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 78. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[20, 21, 22, 23] and target=[120, 121, 122, 123, 124] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(4, 5, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (4, 5, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0078/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 79. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[30, 31, 32, 33, 34] and target=[130, 131] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(5, 2, Yes, no_injection_finite_pigeonhole)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (5, 2, Yes, no_injection_finite_pigeonhole).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0079/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 80. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[40, 41] and target=[140, 141, 142] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(2, 3, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (2, 3, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0080/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 81. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[50, 51, 52] and target=[150, 151, 152, 153] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(3, 4, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (3, 4, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0081/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 82. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[60, 61, 62, 63] and target=[160, 161, 162, 163, 164] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(4, 5, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (4, 5, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0082/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 83. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[70, 71, 72, 73, 74] and target=[170, 171] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(5, 2, Yes, no_injection_finite_pigeonhole)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (5, 2, Yes, no_injection_finite_pigeonhole).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0083/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 84. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[80, 81] and target=[180, 181, 182] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(2, 3, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (2, 3, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0084/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 85. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[90, 91, 92] and target=[190, 191, 192, 193] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(3, 4, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (3, 4, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0085/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 86. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[100, 101, 102, 103] and target=[200, 201, 202, 203, 204] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(4, 5, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (4, 5, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0086/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 87. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[110, 111, 112, 113, 114] and target=[210, 211] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(5, 2, Yes, no_injection_finite_pigeonhole)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (5, 2, Yes, no_injection_finite_pigeonhole).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0087/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 88. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[120, 121] and target=[220, 221, 222] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(2, 3, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (2, 3, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0088/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 89. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[130, 131, 132] and target=[230, 231, 232, 233] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(3, 4, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (3, 4, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0089/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 90. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[140, 141, 142, 143] and target=[240, 241, 242, 243, 244] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(4, 5, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (4, 5, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0090/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 91. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[150, 151, 152, 153, 154] and target=[250, 251] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(5, 2, Yes, no_injection_finite_pigeonhole)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (5, 2, Yes, no_injection_finite_pigeonhole).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0091/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 92. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[160, 161] and target=[260, 261, 262] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(2, 3, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (2, 3, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0092/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 93. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[170, 171, 172] and target=[270, 271, 272, 273] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(3, 4, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (3, 4, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0093/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 94. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[180, 181, 182, 183] and target=[280, 281, 282, 283, 284] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(4, 5, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (4, 5, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0094/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 95. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[190, 191, 192, 193, 194] and target=[290, 291] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(5, 2, Yes, no_injection_finite_pigeonhole)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (5, 2, Yes, no_injection_finite_pigeonhole).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0095/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 96. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[200, 201] and target=[300, 301, 302] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(2, 3, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (2, 3, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0096/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 97. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[210, 211, 212] and target=[310, 311, 312, 313] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(3, 4, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (3, 4, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0097/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 98. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[220, 221, 222, 223] and target=[320, 321, 322, 323, 324] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(4, 5, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (4, 5, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0098/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 99. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[230, 231, 232, 233, 234] and target=[330, 331] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(5, 2, Yes, no_injection_finite_pigeonhole)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (5, 2, Yes, no_injection_finite_pigeonhole).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0099/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate





## 100. A Larger Finite Source Rules Out Every Injection

Use the complete finite source=[240, 241] and target=[340, 341, 342] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.

View answer and explanation(2, 3, No, no_impossibility_certificate)

Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (2, 3, No, no_impossibility_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0100/)

Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261030-r05-cardinality-comparison-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261030-r05-cardinality-comparison-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261030-r05-cardinality-comparison-foundations/validation-report.json)