Definition:Disjoint Union (Set Theory)

Definition
Let $\family {S_i}_{i \mathop \in I}$ be an $I$-indexed family of sets.

The disjoint union of $\family {S_i}_{i \mathop \in I}$ is defined as the set:
 * $\ds \bigsqcup_{i \mathop \in I} S_i = \bigcup_{i \mathop \in I} \set {\tuple {x, i}: x \in S_i}$

where $\bigcup$ denotes union.

Each of the sets $S_i$ is canonically embedded in the disjoint union as the set:
 * ${S_i}^* = \set {\tuple {x, i}: x \in S_i}$

For distinct $i, j \in I$, the sets ${S_i}^*$ and ${S_j}^*$ are disjoint even if $S_i$ and $S_j$ are not.

If $S$ is a set such that $\forall i \in I: S_i = S$, then the disjoint union (as defined above) is equal to the cartesian product of $S$ and $I$:
 * $\ds \bigsqcup_{i \mathop \in I} S = S \times I$

Also see

 * Definition:Sum of Cardinals