Disjoint union
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In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint.
Formally, if <math>C<math> is a collection of sets, then
- <math>\bigcup_{A \in C} A<math>
is a disjoint union if and only if for all A and B in C
- <math>A \neq B \implies A \cap B = \varnothing.<math>
The term disjoint union is also used to refer to a modified union operation which indexes the elements according to which set they originated in, ensuring that the result is a disjoint union in the above sense. This allows one to take the disjoint union of a collection of sets that are not in fact disjoint.
Formally, let {Ai : i ∈ I} be a family of sets indexed by I. The disjoint union of this family is the set
- <math>\coprod_{i\in I}A_i = \bigcup_{i\in I}\{(x,i) : x \in A_i\}<math>
The elements of the disjoint union are ordered pairs (x, i). Here i serves as an auxiliary index that indicates which Ai the element x came from. Note that each of the sets Ai can be canonically embedded in the disjoint union as the set
- <math>A_i^* = \{(x,i) : x \in A_i\}<math>
Observe that for i ≠ j, the sets Ai* and Aj* are disjoint even if the sets Ai and Aj are not.
Consider the extreme case where each of the Ai are equal to some fixed set A for each i ∈ I. In this case one can show that the disjoint union of this family is the Cartesian product of A and I:
- <math>\coprod_{i\in I}A_i = A \times I.<math>
One may occasionally see the notation
- <math>\sum_{i\in I}A_i<math>
for the disjoint union of a family of sets, or the notation A + B for the disjoint union of two sets. 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. See coproduct for more details.