disjoint union

Disjoint Union

The disjoint union $A\bigsqcup B$ of two sets $A$ and $B$ is the set containing all elements of $A$ and $B$, but with elements tagged by their origin, so that even if $A$ and $B$ have elements in common, they are still distinct in the disjoint union.

Formally, let $(A_i:i\in I)$ be an indexed family of sets. The disjoint union ${⨆}_{i\in I}{A}_i$ is defined as:

$$\underset{i\in I}{⨆}{A}_i=\bigcup _{i\in I}(A_i\times \{i\})=\{(x,i):x\in A_i,i\in I\}$$

The cartesian product of the set $A_i$ with the singleton set $\{i\}$ tags each element of $A_i$ with its index $i$.

If $E$ is the disjoint union of $\{A_j\}$, then $\{A_j\}$ is a partition of $E$.