Intersection

Intersection

A set intersection is a binary operation, which means two sets are required to perform this operation. The intersection of sets $A$ and $B$ is set $C$, which contains every element that is a member of both $A$ and $B$. We can write this definition as follows:

$$C=A \cap B = \{x \mid x\in A \land x\in B\}$$

If an object is a member of both $A$ and $B$, it will appear in $C$.

The following facts are obvious from the definition of the intersection operation, where $\mathbb{U}$ is the universal set, $A$ is an arbitrary set, and $\emptyset$ is the empty set:

$$A\cap \emptyset = \emptyset \cap A = \emptyset \\A\cap \mathbb{U} = \mathbb{U} \cap A = A$$

Cardinality

Further, if $C=A∩B$, where $A$ and $B$ are finite sets, the cardinality of $C$ cannot be greater than the cardinality of $A$ or $B$—that is, $|C|\le |A|$ and $|C|\le |B|$.