# Set operations

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

Union, intersection, difference, and relative complement construct new sets from existing sets through element-membership conditions.

mko-set-operations · definition · v0.10.0



## Mathematical statement

For sets A and B in one domain, the union contains elements belonging to at least one set, the intersection contains elements belonging to both, and A\B contains elements belonging to A but not B. A complement must be relative to an explicit universe U and is written U\A.





## Representative formula (exact TeX)

A\cup B





## Assumptions


- A and B are sets in the same ambient type or the same explicit universe U.
- When complement is used, the universe U or ambient type is explicitly specified.





## Symbol scope


- A — left set
- B — right set
- ∪ — set-union operation





## Intuition, definition, and misconceptions

A∪B contains every element appearing in A or B, with repeated occurrences still counting as one set element. A∩B retains only elements common to both sets.

A\B is directional and retains only elements in A that are not in B, so it generally differs from B\A.

Complement is not an absolute operation detached from a domain. A finite exercise must first list its universe U, after which the relative complement of A is U\A. In Lean Set α, complement is interpreted relative to the ambient type α and its universal set.

Common mistakes include swapping union and intersection, assuming A\B equals B\A, or listing a complement without specifying a universe.





## Adopted derivation guide

[Set operations: union, intersection, difference and complement in an explicit universe](https://ocme.evemisslab.com/en/derivations/derivation-mko-set-operations-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Set membership](https://ocme.evemisslab.com/en/mko/mko-set-membership/) — All four set operations define their output through membership in the input sets.





## Verification boundary

This entry is an AI candidate for set operations. Finite enumeration checks only concrete examples and is not a proof of universal properties of the four operations.

not_formalized

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





## Next obligation

Bind exact statements, sources, versions, and axiom receipts for Set.mem_union, intersection membership, Set.mem_sdiff, and Set.mem_compl_iff separately.





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