Disjoint Union (Set)

Definition

Disjoint Union

The disjoint union of sets $A$ and $B$ combines their elements while preserving which set each came from:

$$A\uplus B:=(A\times \{0\})\cup (B\times \{1\}).$$

The tags $0,1$ distinguish the two sources. If $x\in A\cap B$, its copies $(x,0)$ and $(x,1)$ remain distinct. Thus $|A\uplus B|=|A|+|B|$ for finite sets, even when $A$ and $B$ overlap. The notation $A\bigsqcup B$ is also common.