# Question 14808: The Last Equality Flag Is a Bijection-Supported Certificate [Return to catalogue page 75](https://ocme.evemisslab.com/en/questions/page/75/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261030-r05-cardinality-comparison-foundations/#q-20261014-advance-20261030-r05-cardinality-comparison-0058) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0028-14808 · q-20261014-advance-20261030-r05-cardinality-comparison-0058 ## Original question and givens 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. - G001 — 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. ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-cardinality-comparison](https://ocme.evemisslab.com/en/mko/mko-cardinality-comparison/) ## Step-by-step derivation S001 · impliesBefore exact ordered source=[70, 71, 72, 73, 74];exact ordered target=[170, 171];aligned mapping=[170, 171, 170, 171, 170] After total typed map on these distinct source/target lists;orders retained Conditions: none A supplied mapping binds distinct source/target lists in original order,with exactly one permitted image for each source element.Length comparison alone is not mapping qualification. Step identifiers and dependenciesRule: cardinality_exact_bound_total_mapping · Uses: G001 S002 · impliesBefore distinct source indices=[0, 1];source labels=[70, 71];images=[170, 171] After same image=No Conditions: none The source elements are distinct;compare images for every pair.An empty source is vacuously injective,without automatically being surjective. Step identifiers and dependenciesRule: cardinality_every_source_pair_collision_check · Uses: G001, S001 S003 · impliesBefore distinct source indices=[0, 2];source labels=[70, 72];images=[170, 170] After same image=Yes Conditions: none The source elements are distinct;compare images for every pair.An empty source is vacuously injective,without automatically being surjective. Step identifiers and dependenciesRule: cardinality_every_source_pair_collision_check · Uses: G001, S002 S004 · impliesBefore distinct source indices=[0, 3];source labels=[70, 73];images=[170, 171] After same image=No Conditions: none The source elements are distinct;compare images for every pair.An empty source is vacuously injective,without automatically being surjective. Step identifiers and dependenciesRule: cardinality_every_source_pair_collision_check · Uses: G001, S003 S005 · impliesBefore distinct source indices=[0, 4];source labels=[70, 74];images=[170, 170] After same image=Yes Conditions: none The source elements are distinct;compare images for every pair.An empty source is vacuously injective,without automatically being surjective. Step identifiers and dependenciesRule: cardinality_every_source_pair_collision_check · Uses: G001, S004 S006 · impliesBefore distinct source indices=[1, 2];source labels=[71, 72];images=[171, 170] After same image=No Conditions: none The source elements are distinct;compare images for every pair.An empty source is vacuously injective,without automatically being surjective. Step identifiers and dependenciesRule: cardinality_every_source_pair_collision_check · Uses: G001, S005 S007 · impliesBefore distinct source indices=[1, 3];source labels=[71, 73];images=[171, 171] After same image=Yes Conditions: none The source elements are distinct;compare images for every pair.An empty source is vacuously injective,without automatically being surjective. Step identifiers and dependenciesRule: cardinality_every_source_pair_collision_check · Uses: G001, S006 S008 · impliesBefore distinct source indices=[1, 4];source labels=[71, 74];images=[171, 170] After same image=No Conditions: none The source elements are distinct;compare images for every pair.An empty source is vacuously injective,without automatically being surjective. Step identifiers and dependenciesRule: cardinality_every_source_pair_collision_check · Uses: G001, S007 S009 · impliesBefore distinct source indices=[2, 3];source labels=[72, 73];images=[170, 171] After same image=No Conditions: none The source elements are distinct;compare images for every pair.An empty source is vacuously injective,without automatically being surjective. Step identifiers and dependenciesRule: cardinality_every_source_pair_collision_check · Uses: G001, S008 S010 · impliesBefore distinct source indices=[2, 4];source labels=[72, 74];images=[170, 170] After same image=Yes Conditions: none The source elements are distinct;compare images for every pair.An empty source is vacuously injective,without automatically being surjective. Step identifiers and dependenciesRule: cardinality_every_source_pair_collision_check · Uses: G001, S009 S011 · impliesBefore distinct source indices=[3, 4];source labels=[73, 74];images=[171, 170] After same image=No Conditions: none The source elements are distinct;compare images for every pair.An empty source is vacuously injective,without automatically being surjective. Step identifiers and dependenciesRule: cardinality_every_source_pair_collision_check · Uses: G001, S010 S012 · impliesBefore target label=170;exact original mapping=[170, 171, 170, 171, 170] After preimage source indices=[0, 2, 4];covered=Yes Conditions: none Every target element needs a nonempty preimage for surjectivity.Record original source indices,not image-as-index values.Coverage of an empty target is vacuous,but total-map typedness still must qualify first. Step identifiers and dependenciesRule: cardinality_every_target_fiber_coverage · Uses: G001, S011 S013 · impliesBefore target label=171;exact original mapping=[170, 171, 170, 171, 170] After preimage source indices=[1, 3];covered=Yes Conditions: none Every target element needs a nonempty preimage for surjectivity.Record original source indices,not image-as-index values.Coverage of an empty target is vacuous,but total-map typedness still must qualify first. Step identifiers and dependenciesRule: cardinality_every_target_fiber_coverage · Uses: G001, S012 S014 · impliesBefore injective=No;surjective=Yes;source size=5;target size=2 After bijective=No;bijection-supported equality certificate=No Conditions: none The source protocol gates the last field on this map's bijectivity.A bad map between equal-sized sets still returns No:it supplied no equality certificate,not a claim that the sizes differ.The unique map between two empty sets is bijective. Step identifiers and dependenciesRule: bijection_equality_flag_not_bare_size_equality · Uses: G001, S013 ## Reverse checks - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([70, 71, 72, 73, 74], [170, 171], [170, 171, 170, 171, 170], No, Yes, No, No) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 8dde330393a8659c08e88439b02af7a3d6f7a9ad59be9005aac16e6f0e5b3496 - candidate record SHA-256: 9bf8c284a7817b6e09f39e9015eb9c77b97cd6b02380e8f712b194927f4bfe68 - semantic review receipt SHA-256: bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209ae - question-first witness SHA-256: 61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261030-r05-cardinality-comparison-0058.json)