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Question 2808: Existence of a Bijection from Finite Cardinalities

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0004-02808 · q-20261004-advance-20261005-r03-bijective-0083

Original question and givens

The explicit finite cardinalities are |A|=3 and |B|=4, with no particular function table supplied. Does some bijection A→B exist?

  • G001 — The explicit finite cardinalities are |A|=3 and |B|=4, 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.

Step-by-step derivation

S001 · implies

Before

finite |A|=3; |B|=4

After

equal finite cardinalities=No

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 dependencies

Rule: finite_bijection_cardinality · Uses: G001

S002 · implies

Before

m=3; n=4

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 dependencies

Rule: finite_pairing_construction · Uses: G001, S001

S003 · implies

Before

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 dependencies

Rule: 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

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: 65cea4233cf9d07f9352ade72525c171f26fcc5c88495eeddc9680a3e0af462d
  • candidate record SHA-256: c7adaa4751b9e3dd1270201b1f59fb5a55dcd44cc929f05424941c84f3d2a84c
  • semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56
  • question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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