Definition:Disjoint Union (Set Theory)

From ProofWiki

Jump to: navigation, search

Definition

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

The disjoint union of $\mathcal A$ is defined as the set:

$\displaystyle \bigsqcup_{i \in I} A_i = \bigcup_{i \in I} \left\{{\left({x,i}\right) : x \in A_i}\right\}$

The elements of the disjoint union are ordered pairs $\left({x, i}\right)$, where:

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 $I$:

$\displaystyle \bigsqcup_{i \in I} A = A \times I$

This is also called a discriminated union.

Occasionally the notation:

$\displaystyle \sum_{i \in I} A_i$

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.

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