The intersection of two sets $A$ and $B$ is the set that contains all the elements $x$ such that $x \in A$ and $x \in B$ . The intersection of $A$ and $B$ is written as $A \cap B$ . The following Venn diagram illustrates the intersection of two sets $A$ and $B$ :

Example. If $A=\{1,2,3,4,5\}$ and $B=\{1,3,5,7,9\}$ then $A\cap B=\{1,3,5\}$ .

We can also define the intersection of an arbitrary number of sets. If $\{A_j\}_{j\in J}$ is a family of sets, we define the intersection of all them, denoted $\bigcap_{j\in J} A_j$ , as the set consisting of those elements belonging to every set $A_j$ : $$ \bigcap_{j\in J} A_ j = \{x: x\in A_j \mbox{ for all } j\in J \}. $$

A set $U$ intersects, or meets, a set $V$ if $U\cap V$ is non-empty.

Some elementary properties of $\cap$ are

• (idempotency) $A\cap A = A$ ,
• (commutativity) $A\cap B=B\cap A$ ,
• (associativity) $A\cap (B\cap C) = (A\cap B)\cap C$ ,
• $A\cap A^\complement = \varnothing$ , where $A^\complement$ is the complement of $A$ in some fixed universe $U$ .

Remark. What is $\bigcap_{j\in J} A_j$ when $J=\varnothing$ ? In other words, what is the intersection of an empty family of sets? First note that if $I\subseteq J$ , then $$\bigcap_{j\in J} A_j \subseteq \bigcap_{i\in I} A_i.$$ This leads the conclusion that the intersection of an empty family of sets should be as large as possible. How large should it be? In addition, is this intersection a set? The answer depends on what versions of set theory we are working in. Some theories (for example, von Neumann-Gödel-Bernays) say this is the class $V$ of all sets, while others do not define this notion at all. However, if there is a fixed set $U$ in advance such that each $A_j\subseteq U$ , then it is sometimes a matter of convenience to define the intersection of an empty family of $A_j$ to be $U$ .