\begin{definition}[Definition:Set Intersection]
Let $S$ and $T$ be sets.
The '''(set) intersection''' of $S$ and $T$ is written $S \cap T$.
It means the set which consists of all the elements which are contained in both of $S$ and $T$:
:$x \in S \cap T \iff x \in S \land x \in T$
or, more formally:
:$A = S \cap T \iff \forall z: \paren {z \in A \iff z \in S \land z \in T}$
We can write:
:$S \cap T := \set {x: x \in S \land x \in T}$
and can voice it '''$S$ intersect $T$'''.
It can be seen that, in this form, $\cap$ is a binary operation which acts on sets.
One often says that two sets '''intersect''' {{iff}} they have non-empty '''intersection'''.
\end{definition}