Definition:Set Intersection

Definition
Let $$S$$ and $$T$$ be any two sets.

The (set) intersection of $$S$$ and $$T$$ is written $$S \cap T$$ and means the set which consists of all the elements which are contained in both of $$S$$ and $$T$$:


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

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 $$\mathbb{S}$$.