Category:Definitions/Disjoint Unions

This category contains definitions related to Disjoint Union in the context of Set Theory.
Related results can be found in Category:Disjoint Unions.

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:

${S_i}^* = \set {\tuple {x, i}: x \in S_i}$