The union of a set $X$ is
\[\bigcup X = \{x : (\exists Y \in X : x \in Y)\}.\]This is a set by Axiom of Union.
For any sets $A$, $B$, $C$, $D$, etc., we write
\[\begin{align*} A \cup B &= \bigcup \{A, B\}, \\ A \cup B \cup C &= (A \cup B) \cup C, \\ A \cup B \cup C \cup D &= (A \cup B \cup C) \cup D, \\ &\text{etc.} \end{align*}\]
- \[\bigcup \empt = \empt.\]
For any set $X$,
\[X \cup \empt = X.\]
For any set $X$,
\[X \cup X = X.\]
For any sets $X$ and $Y$,
\[X \cup Y = Y \cup X.\]
For any sets $X$, $Y$ and $Z$,
\[(X \cup Y) \cup Z = X \cup (Y \cup Z).\]
As a corollary of Proposition 2 (II), commutativity and associativity of $\cup$:
For any set $X$, $(\powerset(X), \cup)$ is an abelian monoid with neutral element $\empt$.