\begin{definition}[Definition:Disjoint Union (Set Theory)/2 Sets]
Let $I$ be a doubleton, say $I := \set {0, 1}$.
Let $\family {S_i}_{i \mathop \in I}$ be an $I$-indexed family of sets.
Then the '''disjoint union''' of $\family {S_i}_{i \mathop \in I}$ is defined as the set:
{{begin-eqn}}
{{eqn | l = \bigsqcup_{i \mathop \in I} S_i
      | r = \bigcup_{i \mathop \in I} \set {\tuple {x, 0}, \tuple {y, 1} : x \in S_0, y \in S_1}
      | c = 
}}
{{eqn | r = \bigcup \set {S_0 \times \set 0, S_1 \times \set 1}
      | c = 
}}
{{eqn | r = \paren {S_0 \times \set 0} \cup \paren {S_1 \times \set 1}
      | c = 
}}
{{eqn | r = S_0 \sqcup S_1
      | c = 
}}
{{end-eqn}}
\end{definition}