> [!NOTE] Definition (Intersection) > Let $X$ be a [[Sets|set]] of sets. Then the intersection of $X$ is given by $\bigcap_{A\in X} A = \{ z \mid z \in A \text{ for all } A\in X \}$ # Properties > [!NOTE] Lemma (Intersection Properties) > Suppose that $X, Y$, and $Z$ are sets. Then we have the following. > (1) $X \cap \varnothing=\varnothing$ (annihilator) > (2) $(X \cap Y) \cap Z=X \cap(Y \cap Z)$ (associativity) > (3) $X \cap Y=Y \cap X$ (commutativity) > (4) $X \subset Y$ if and only if $X \cap Y=X$ (absorption) > (5) $X \cap X=X$ (idempotent) >Proof. By [[De Morgan's Laws for Intersection]], ...