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:


 * $A = S \cap T \iff \forall z: \paren {z \in A \iff z \in S \land z \in T}$

We can write:


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

It can be seen that, in this form, $\cap$ is a binary operation which acts on sets.

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

Also known as
The intersection of sets is also known as the product, but this is usually considered old-fashioned nowadays.

The term meet can also be seen, but this is usually reserved for meet in order theory.

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 $\displaystyle \Bbb S = \set S \implies \bigcap \Bbb S = S$
 * Intersection of Empty Set, where it is shown (paradoxically) that $\displaystyle \Bbb S = \set \O \implies \bigcap \Bbb S = \Bbb U$