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.

Generalized Notation
Let $I$ be an indexing set.

Let $\left \langle {X_i} \right \rangle_{i \in I}$ be a family of subsets of a set $S$.

Then the intersection of $\left \langle {X_i} \right \rangle$ is defined as:


 * $\displaystyle \bigcap_{i \in I} X_i = \left\{{y: \forall i \in I: y \in X_i}\right\}$

This notation can also be used as $\displaystyle \bigcap_i X_i$ to be written $\displaystyle \bigcap_{i \in I} X_i$.

The indexing set itself can be disposed of, as follows:

If $\Bbb S$ is a set of sets, then the intersection of $\Bbb S$ is:
 * $\displaystyle \bigcap \Bbb S = \left\{{x: \forall S \in \Bbb S: x \in S}\right\}$

That is, the set of all objects that are elements of all the elements of $\Bbb S$.

Thus:
 * $\displaystyle S \cap T = \bigcap \left\{{S, T}\right\}$

Countable Intersection
Let $S = S_1 \cap S_2 \cap \ldots \cap S_n$. Then:


 * $\displaystyle \bigcap_{i \in \N^*_n} S_i = \left\{{x: \forall i \in \N^*_n: x \in S_i}\right\}$

If it is clear from the context that $i \in \N^*_n$, we can also write $\displaystyle \bigcap_{\N^*_n} S_i$.

An alternative notation for the same concept is $\displaystyle \bigcap_{i=1}^n S_i$.

If $\Bbb S$ is a set of sets, then the intersection of $\Bbb S$ is:
 * $\displaystyle \bigcap \Bbb S = \left\{{x: \forall S \in \Bbb S: x \in S}\right\}$

That is, the set of all objects that are elements of all the elements of $\Bbb S$.

Thus:
 * $\displaystyle S \cap T = \bigcap \left\{{S, T}\right\}$

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


 * VennDiagramSetIntersection.png

Also see

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