# Bijectivity and unique preimages

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

A bijection is both injective and surjective; equivalently, every codomain element has exactly one permitted preimage.

mko-bijective · definition · v0.10.0



## Mathematical statement

A total function f:A→B is bijective exactly when it is both injective and surjective, or equivalently when each fiber of b∈B has exactly one element. For finite sets, some bijection A→B exists exactly when |A|=|B|; equal cardinality does not make every specified f bijective.





## Representative formula (exact TeX)

f:A\to B





## Assumptions


- Testing a given function and asking whether some bijection exists are distinct questions.
- Cardinality-existence exercises use finite sets with nonnegative integer sizes; size zero on both sides permits the unique empty bijection.





## Symbol scope


- f — total function to be checked
- A — domain
- B — codomain





## Intuition, definition, and misconceptions

A finite function can be bijective, injective only, surjective only, or neither. Checking set sizes alone omits the actual mapping data.

A complete fiber profile lists all codomain targets in order, including zero-size fibers. All fibers must have size one for a bijection.

Equal finite cardinalities allow pairing to construct some bijection, but a different given function may fail to be bijective. A cardinality exercise asks only about existence and does not adjudicate a supplied function.

Common errors include inferring bijectivity of a given f directly from |A|=|B|, or checking only one of the two properties.





## Adopted derivation guide

[Bijectivity: testing given functions by fiber profiles, kept apart from whether some bijection exists](https://ocme.evemisslab.com/en/derivations/derivation-mko-bijective-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Injectivity and distinct inputs](https://ocme.evemisslab.com/en/mko/mko-injective/) — Bijectivity includes the injectivity condition.
- [Surjectivity and codomain coverage](https://ocme.evemisslab.com/en/mko/mko-surjective/) — Bijectivity includes the surjectivity condition.





## Verification boundary

Finite fibers and cardinalities can be recomputed; general bijection, equinumerosity, and inverse-existence results have no attached formal Evidence.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Bind Function.Bijective, existsUnique, and the finite bridge through Fintype.card and Equiv.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/bijective.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-bijective/index.md)