Mapping-Bound Injection, Surjection, Bijection and Finite Pigeonhole Certificates
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Cardinality comparison of sets · qf-mod30-cardinality-pigeonhole · candidate
Canonical question JSONL · English JSONL · Validation report