Intersection of sets

Definition · applies to sets · also called $\cap$

High school

Consider two sets $A$ and $B$. The intersection of $A$ and $B$, written $A \cap B$, is the set of every element that is in both $A$ and $B$.

Undergraduate

Consider two sets $A$ and $B$. The intersection of sets $A$ and $B$ is

\[A \cap B = \setdef{x \suchthat x \in A \text{ and } x \in B}.\]

Examples

$A = \setdef{1, 2}$ and $B = \setdef{3, 4}$ : $A \cap B = \emptyset$ Elementary
$A$ and $B$ have no elements in common, so their intersection is empty, and removing $B$ from $A$ changes nothing.
$A = \setdef{\text{apple}, \text{orange}}$ and $B = \setdef{\text{orange}, \text{banana}}$ : $A \cap B = \setdef{\text{orange}}$ High school

Sets $A$ and $B$ are disjoint if $A \cap B = \emptyset$.