So, an element belonging to both A and B appears twice in the disjoint union, with two different labels.

A disjoint union of an indexed family of sets

such that the images of these injections form a partition of

A disjoint union of a family of pairwise disjoint sets is their union.

In category theory, the disjoint union is the coproduct of the category of sets, and thus defined up to a bijection.

The disjoint union of two sets

is written with infix notation as

Some authors use the alternative notation

A standard way for building the disjoint union is to define

as the set of ordered pairs

where the second element in each pair matches the subscript of the origin set (for example, the

The disjoint union of this family is the set

The elements of the disjoint union are ordered pairs

serves as an auxiliary index that indicates which

is canonically isomorphic to the set

is canonically embedded in the disjoint union.

is equal to some fixed set

the disjoint union is the Cartesian product of

is used for the disjoint union of a family of sets, or the notation

This notation is meant to be suggestive of the fact that the cardinality of the disjoint union is the sum of the cardinalities of the terms in the family.

Compare this to the notation for the Cartesian product of a family of sets.

In the language of category theory, the disjoint union is the coproduct in the category of sets.

It therefore satisfies the associated universal property.

This also means that the disjoint union is the categorical dual of the Cartesian product construction.

For many purposes, the particular choice of auxiliary index is unimportant, and in a simplifying abuse of notation, the indexed family can be treated simply as a collection of sets.

In category theory the disjoint union is defined as a coproduct in the category of sets.

As such, the disjoint union is defined up to an isomorphism, and the above definition is just one realization of the coproduct, among others.

When the sets are pairwise disjoint, the usual union is another realization of the coproduct.

This justifies the second definition in the lead.

This categorical aspect of the disjoint union explains why