Subsetthat

Subsetthat is a term used in set theory and related disciplines to denote a subcollection of the power set of a universal set U that is defined by a predicate P on subsets of U. For a given U and a predicate P: P takes a subset S ⊆ U and returns true or false. The subsetthat determined by P is the family F = { S ⊆ U | P(S) is true }. This construction generalizes many common subset families, such as the nonempty subsets or subsets meeting a specified condition.

Formally, if P is a property of subsets of U, then F is the set of all S ⊆ U satisfying P. Examples: P(S) = "S contains a fixed element a" yields all subsets containing a; P(S) = "S has even cardinality" yields all even-sized subsets. If P is monotone with respect to inclusion (S ⊆ T implies P(S) implies P(T)), then F is upward closed: S ∈ F and S ⊆ T ⊆ U implies T ∈ F. In that case F is closed under unions; closure under intersections depends on P’s additional behavior.

Subsetthat provides a unifying lens for discussions of definable families and constraint-based constructions in combinatorics, database theory, and knowledge representation. Its computational aspects center on evaluating P on candidate subsets, which may range from simple to combinatorially expensive.

See also: definable set families, filters, ideals, hypergraphs. The term subsetthat is not standardized, but appears in some informal treatments to describe families of subsets defined by a shared predicate.