Groupsfrom

Groupsfrom is a term used to describe the operation of forming groups from a collection of items based on a specified grouping criterion. It is used across mathematics, statistics, and data processing to organize elements into manageable subsets.

Formally, let X be a set and f: X -> A be a grouping function that assigns to each element a label in A. The groupsfrom operation produces the family { X_a : a in Im(f) } where X_a = { x in X | f(x) = a }. If f induces an equivalence relation on X, the resulting groups are the equivalence classes and partition X.

Distinctions: when each element receives exactly one label, the groups are disjoint and cover X; with multi-valued criteria, elements may appear in multiple groups.

Examples: grouping customers by country, items by category, or documents by topic tag. In programming and databases, group by operations implement groupsfrom to compute aggregates per group.

Relation to other concepts: closely related to partition, equivalence relation, and clustering; grouping differs from clustering in that the former uses predefined keys, while clustering seeks to discover structure. In combinatorics, groupsfrom can be used to study partitions of a set and related counting problems.

See also: partition, equivalence relation, grouping, clustering, group by (computing).