Unioncombination

Unioncombination is a concept in combinatorics and set theory that describes creating a new family of sets by taking unions of chosen members from several input families. It provides a way to build complex sets from simpler components by combining their elements through set union.

Formally, let F1, F2, ..., Fk be nonempty families of subsets of a universe U. The unioncombination of these families is the new family

{ A1 ∪ A2 ∪ ... ∪ Ak : for each i, Ai ∈ Fi }.

This operation yields a collection of subsets of U whose members are obtained by selecting one set from each Fi and taking their union. The result is independent of the order of the inputs, so permuting the Fi does not change the produced family.

Example: If F1 = {{1}, {2}} and F2 = {{a}, {b}}, then the unioncombination is

{ {1} ∪ {a}, {1} ∪ {b}, {2} ∪ {a}, {2} ∪ {b} } = { {1, a}, {1, b}, {2, a}, {2, b} }.

Properties and notes:

- The size of the unioncombination is at most the product of the sizes of the input families: |UnionComb(F1,...,Fk)| ≤ ∏i |Fi|.

- The operation yields a new family of subsets of the same universe and can be used to study how complex sets arise from simpler building blocks.

- It relates to other constructions such as Cartesian products and power set operations, but focuses on unions of selected members rather than combinations of elements.

Applications include constructing test collections in computer science, modeling combined events in probability, and analyzing union-closed families in extremal set theory.

See also: Cartesian product, power set, union-closed family, combinatorial design, set operations.