Definition:Set Intersection

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

The 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 $$T$$.

$$x \in S \cap T \iff x \in S \land x \in T$$

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:

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

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

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

Lines and Geometric Figures
The intersection of two lines $$AB$$ and $$CD$$ is denoted by $$AB\cap CD$$. The intersection of two geometric figures is the set of points shared by both figures.