The union of $X$, written as
\[\bigcup X,\]is the unique set $Y$ such that
\[\forall y :\enspace y\in Y \enspace\lrimp\enspace \exists x:x\in X\land y\in x.\]The existence of $Y$ is justified by Axiom of Union and its uniqueness is justified by Axiom of Extensionality.
We write
\[\begin{align*} A\cup B &= \bigcup\{A,B\}, \\ A\cup B\cup C &= (A\cup B)\cup C, \\ A\cup B\cup C\cup D &= (A\cup B\cup C)\cup D \end{align*}\]and so on.
- \[\bigcup\varnothing = \varnothing.\]
For any $X$,
\[X\cup\varnothing = X.\]
For any $X$,
\[X\cup X = X.\]
For any $X$ and $Y$,
\[X\cup Y = Y\cup X.\]
For any $X$, $Y$ and $Z$,
\[(X\cup Y)\cup Z = X\cup(Y\cup Z).\]
$(\V,\cup)$ is an abelian monoid with neutral element $\varnothing$.
Proof.By commutativity, associativity of $\cup$ and U#PROP-EMP (II).