# Cardinality comparison of sets

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

An injection, surjection or bijection between explicit finite source and target sets supplies a corresponding finite cardinality-comparison certificate.

mko-cardinality-comparison · concept · v0.10.0



## Mathematical statement

This slice is limited to complete finite sets. An injection A→B certifies |A|≤|B|, a surjection certifies |A|≥|B|, and a bijection certifies equality. If |A|>|B| there is no injection. Reverse comparison from a surjection for infinite sets requires additional choice or section obligations and is outside this slice.





## Representative formula (exact TeX)

C(A,B)





## Assumptions


- Source and target are complete lists of distinct elements, and the mapping list has exactly the source length.
- Every image belongs to the target, and the certificate covers only those specified finite sets.





## Symbol scope


- C — cardinality comparison
- A — source set
- B — target set





## Intuition, definition, and misconceptions

First verify a total mapping whose images lie in the target.

Injectivity and surjectivity are separate obligations.

The pigeonhole obstruction is finite-only and is not extended to infinite cardinalities.

Injectivity cannot be discussed without fixing source and target, and the finite surjection comparison cannot be extended unconditionally to infinite sets.





## Adopted derivation guide

[Cardinality certificates from full maps](https://ocme.evemisslab.com/en/derivations/derivation-mko-cardinality-comparison-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/) — An injection supplies one direction of comparison.
- [Surjectivity and codomain coverage](https://ocme.evemisslab.com/en/mko/mko-surjective/) — A surjection supplies the reverse comparison for finite sets.
- [Bijectivity and unique preimages](https://ocme.evemisslab.com/en/mko/mko-bijective/) — A bijection gives equality of cardinalities.
- [Set](https://ocme.evemisslab.com/en/mko/mko-set/) — The mapping is bound to explicit source and target sets.





## Verification boundary

Only explicit finite sets are handled; infinite comparison and choice are outside this slice.

not_formalized

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





## Next obligation

Add at-most-countable sets.





## 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/cardinality-comparison.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-cardinality-comparison/index.md)