Disjoint union

In set theory, a disjoint union is a type of union (Set theoretic union), in which each element of the union is disjoint from the others: intersection with every other element of the union is the empty set.

i.e. Suppose C is a collection of sets, then:

is a disjoint union if and only if

To take the disjoint union of sets that are not in fact disjoint, one can use an indexing device. For example given A₁ and A₂, which may have common elements, with union B, the disjoint union as a subset of B x {1,2} is the union of A₁ x {1} and A₂x{2}.