> [!NOTE] Definition (Union) > Let $X$ be a [[Set|set]] of sets. Then the union of $X$ is given by $\bigcup_{A\in X} A = \{ z \mid z \in A \text{ for some } A\in X \}$ # Properties By [[Union Axiom]], this set exists. > [!NOTE] Lemma (Union Properties) > Suppose that $X, Y$, and $Z$ are sets. Then we have the following. > (1) $X \cup \varnothing=X$ (identity) > (2) $(X \cup Y) \cup Z=X \cup(Y \cup Z)$ (associativity) > (3) $X \cup Y=Y \cup X$ (commutativity) > (4) $X \subset Y$ if and only if $X \cup Y=Y$ (absorption) > (5) $X \cup X=X$ (idempotent) *Proof*. By [[Distributivity of Set Union Over Set Intersection]], .... By [[Distributivity of Set Intersection Over Set Union]], ... By [[De Morgan's Laws for Union]], ...