Disjoint Union

Definition 1. If a set $X$ is disjoint, the union of $X$ is called a disjoint union. The notation

\[A = \bigsqcup X\]

means

\[A = \bigcup X \enspace\land\enspace \text{$X$ is disjoint},\]

and we say $A$ is a partition of $X$.

For any sets $X_1$, $\cdots$, $X_n$ ($n \geq 2$), we write

\[X_1 \sqcup \cdots \sqcup X_n = \bigsqcup \{X_1, \cdots, X_n\}.\]

Proposition 1. For any finite disjoint set $X$,

\[\Big\lvert \bigsqcup X \Big\rvert = \sum_{A \in X} \lvert A \rvert.\]