Disjoint union

In set theory, the disjoint union (or discriminated union) of a family of sets is a modified union operation that indexes the elements according to which set they originated in. Or slightly different from this, the disjoint union of a family of subsets is the usual union of the subsets which are pairwise disjoint – disjoint sets means they have no element in common.

Note that these two concepts are different but strongly related. Moreover, it seems that they are essentially identical to each other in category theory. That is, both are realizations of the coproduct of category of sets.

Disjoint union of sets ${\displaystyle A_{0}}$ = {1, 2, 3} and ${\displaystyle A_{1}}$ = {1, 2} can be computed by finding:

so

Formally, let {A_i : i ∈ I} be a family of sets indexed by i. The disjoint union of this family is the set

The elements of the disjoint union are ordered pairs (x, i). Here i serves as an auxiliary index that indicates which A_i the element x came from.

...
Wikipedia