Union

Definition 1. 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

\[\begin{align*} A \cup B &= \bigcup \{A, B\}, \\ A \cup B \cup C &= (A \cup B) \cup C, \\ A \cup B \cup C \cup D &= (A \cup B \cup C) \cup D, \\ &\text{etc.} \end{align*}\]

Proposition 1.

\[\bigcup \empt = \empt.\]

Proposition 2. For any set $X$,

\[X \cup \empt = X.\]

Proposition 3 (Idempotence). For any set $X$,

\[X \cup X = X.\]

Proposition 4 (Commutativity). For any sets $X$ and $Y$,

\[X \cup Y = Y \cup X.\]

Proposition 5 (Associativity). For any sets $X$, $Y$ and $Z$,

\[(X \cup Y) \cup Z = X \cup (Y \cup Z).\]

Proposition 6. For any set $X$, $(\powerset(X), \cup)$ is an abelian monoid with neutral element $\empt$.