Definition:Disjoint Union (Set Theory)

Definition
Let $\mathcal A = \left\{{A_i : i \in I}\right\}$ be a family of sets where $I$ is an indexing set.

The disjoint union of $\mathcal A$ is defined as the set:
 * $\displaystyle \bigsqcup_{i \mathop \in I} A_i = \bigcup_{i \mathop \in I} \left\{{\left({x, i}\right) : x \in A_i}\right\}$

The elements of the disjoint union are ordered pairs $\left({x, i}\right)$, where:
 * $x$ comes from one of the $A_i$
 * $i$ indicates which $A_i$ the particular $x$ came from.

Each of the sets $A_i$ is canonically embedded in the disjoint union as the set:
 * $A_i^* = \left\{{\left({x, i}\right) : x \in A_i}\right\}$

For $i \ne j$, the sets $A_i^*$ and $A_j^*$ are disjoint even if the sets $A_i$ and $A_j$ are not.

Where each of the $A_i$ are all equal to some set $A$ for each $i \in I$, the disjoint union is the cartesian product of $A$ and $I$:
 * $\displaystyle \bigsqcup_{i \mathop \in I} A = A \times I$

Where $A \cap B = \varnothing$, we can define:
 * $A \sqcup B := A \cup B$

where $A \cup B$ is the union of $A$ and $B$.

Also known as
This is also called a discriminated union.

Occasionally the notation:
 * $\displaystyle \sum_{i \mathop \in I} A_i$

can be seen for the disjoint union of a family of sets.

When two sets are under consideration, the notation:
 * $A \sqcup B$

is usually used.

The notation:
 * $A + B$

is also encountered sometimes.

This notation reflects the fact that, from the corollary to Cardinality of Set Union, the cardinality of the disjoint union is the sum of the cardinalities of the terms in the family.

It is motivated by the notation for a coproduct in category theory, combined with Disjoint Union is Coproduct in Category of Sets.

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

Also see

 * Sum of Cardinals