Intersections

Definition

The Venn diagram displays two sets $X$ and $Y$ with the intersection $X\cap Y$ shaded

Let $X$ and $Y$ be subsets of some universal set  $U$. The intersection of $X$ and $Y$, written $X\cap Y$, is the set of all $x$ in  $U$ which are in both of the sets $X$ and $Y$.
Symbolically, $X\cap Y=\lbrace x\in U:x\in X{\text{ and }}x\in Y\rbrace$.

Examples

Example 1

Determine the intersection of the sets $X=\lbrace 1,2,5,8\rbrace$ and $Y=\lbrace 1,3,6,9\rbrace$.

Solution. By definition, we wish to find the set of all elements which are in both of the sets. The only such element is $1$. Thus, our solution is $X\cap Y=\lbrace 1\rbrace$.

Example 2

Prove that for any sets $X$ and $Y$, $X\cap Y\subseteq X$.

Proof. Let $x\in X\cap Y$. That is, $x\in X$ and $x\in Y$. In particular, since $x\in X$ we have that $X\cap Y\subseteq X$.