Definition:Set Intersection

Definition
Let $S$ and $T$ be any two 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:


 * $A = S \cap T \iff \forall z: \left({z \in A \iff z \in S \land z \in T}\right)$

We can write:


 * $S \cap T := \left\{{x: x \in S \land x \in T}\right\}$

For example, let $S = \left \{{1,2,3}\right\}$ and $T = \left \{{2,3,4}\right\}$. Then $S \cap T = \left \{{2,3}\right\}$.

It can be seen that $\cap$ is an operator.

One often says that two sets intersect iff they have non-empty intersection.

Also denoted as
Some authors use the notation $S \ T$ or $S \cdot T$ for $S \cap T$, but this is non-standard and can be confusing.

Also see

 * Definition:Set Union, a related operation.


 * Intersection of Singleton, where it is shown that $\displaystyle \Bbb S = \left\{{S}\right\} \implies \bigcap \Bbb S = S$
 * Intersection of Empty Set, where it is shown (paradoxically) that $\displaystyle \Bbb S = \left\{{\varnothing}\right\} \implies \bigcap \Bbb S = \Bbb U$