Partition of a set

A partition of a set is a collection of nonempty, pairwise disjoint subsets whose union is the entire set. Each subset is called a block.

Set: A collection of distinct objects, called elements. A partition divides one underlying set into blocks.
Indicator function: A function that assigns 1 to members of a specified set and 0 to all other elements. Indicators can encode whether each element belongs to a given block.
Quotient set: The set of equivalence classes induced by an equivalence relation. A partition's blocks can serve as the elements of a quotient set.
Cover (mathematics): A collection of sets whose union contains a specified set. A cover need not have disjoint members or exclude empty sets.
Subset: A set whose every element belongs to another set. Each block must be a subset of the set being partitioned.
Surjective function: A function whose image equals its codomain. The nonempty fibers of a surjection partition its domain.
Quotient space: A topological space formed by identifying points of another space according to an equivalence relation. Its identifications group points into partition-like equivalence classes.
Disjoint union: A union that retains the identities of its component sets even when elements coincide. Partitions supply disjoint blocks for this union construction.
Union: The set of elements that belong to at least one of a collection of sets. The union of all blocks must recover the entire set.
Fiber (mathematics): For a function and an output, the set of inputs mapped to that output. Fibers collect elements assigned to the same block label.

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