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 $$T$$:


 * $$x \in S \cap T \iff x \in S \and x \in T$$

or, slightly more formally:
 * $$A = S \cap T \iff \forall z: \left({z \in A \iff z \in S \and z \in T}\right)$$

We can write:
 * $$S \cap T = \left\{{x: x \in S \and x \in T}\right\}$$

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.

Generalized Notation
Let $$S = S_1 \cap S_2 \cap \ldots \cap S_n$$. Then:


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

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

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

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

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

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

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