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.

Illustration by Venn Diagram
The red area in the following Venn diagram illustrates $S \cap T$:


 * VennDiagramSetIntersection.png

Historical Note
The symbol $\cap$, informally known as cap, was first used by Hermann Grassmann in Die Ausdehnungslehre from 1844. However, he was using it as a general operation symbol, not specialized for intersection.

It was Giuseppe Peano who took this symbol and used it for intersection, in his 1888 work Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann.

Peano also created the large symbol $\bigcap$ for general intersection of more than two sets. This appeared in his Formulario Mathematico (5th edtion, 1908).

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 $\displaystyle \Bbb S = \left\{{\varnothing}\right\} \implies \bigcap \Bbb S = \Bbb U$