# Question 2819: Existence of a Bijection from Finite Cardinalities

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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.

R0004-02819 · q-20261004-advance-20261005-r03-bijective-0094



## Original question and givens

The explicit finite cardinalities are |A|=9 and |B|=9, with no particular function table supplied. Does some bijection A→B exist?


- G001 — The explicit finite cardinalities are |A|=9 and |B|=9, with no particular function table supplied. Does some bijection A→B exist?





## 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-bijective](https://ocme.evemisslab.com/en/mko/mko-bijective/)





## Step-by-step derivation

S001 · impliesBefore

finite |A|=9; |B|=9

After

equal finite cardinalities=Yes

Conditions: none

A bijection between finite sets exists exactly when their cardinalities agree.A bijection pairs each A-element with a distinct B-element and misses no B-element,so the counts agree.

Step identifiers and dependenciesRule: finite_bijection_cardinality · Uses: G001





S002 · impliesBefore

m=9; n=9

After

equal sizes:pair indexed elements;unequal sizes:no bijection

Conditions: none

For equal sizes,enumerate both finite sets and pair equal indices to construct a bijection.For unequal sizes a complete one-to-one pairing is impossible.The unique empty function from0 to0 is a bijection.

Step identifiers and dependenciesRule: finite_pairing_construction · Uses: G001, S001





S003 · impliesBefore

existence of some map,without any supplied table

After

not a verdict about an unspecified particular function

Conditions: none

Equal cardinalities do not make every particular function bijective;this task asks only whether some bijection exists.

Step identifiers and dependenciesRule: existence_vs_given_function · Uses: G001, S002









## Reverse checks


- 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

Yes

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: 44cea468c5dc5e95eee6b39b36f671245d9b86539ce784939b0cedb2647ef31b
- candidate record SHA-256: 6d2ea27d5b240fc34cdbdac495be33272d9c318bd42e6a976d66b68bdaf6f04f
- semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56
- question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261004-advance-20261005-r03-bijective-0094.json)