Intersection

Definition 1. The intersection of a set $X$ is

\[\bigcap X = \left\{ x : (\forall A \in X : x \in A) \right\}.\]

This is a set by Separation Schema: $\bigcap X \subseteq A$ for any $A \in X$.

For any sets $X_1$, $\cdots$, $X_n$ ($n \geq 2$), we write

\[X_1 \cap \cdots \cap X_n = \bigcap \{X_1, \cdots, X_n\}.\]

Proposition 1.

\[\bigcap \empt = \V.\]

Proposition 2. For any set $X$,

\[X \cap \empt = \empt.\]

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

\[X \cap X = X.\]

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

\[X \cap Y = Y \cap X.\]

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

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

Proposition 6. For any set $X$, $(\powerset(X), \cap)$ is an abelian semigroup.