Intersection

set-theory

Definition

Intersection

Let $U$ be a universe and let $A,B\subseteq U$. The intersection of $A$ and $B$ is the set

$$A\cap B=\{x\in U\mid x\in A\wedge x\in B\}.$$

More generally, for a family $(A_i{)}_{i\in I}$ of subsets of $U$,

$$\bigcap _{i\in I}{A}_i=\{x\in U\mid \forall i\in I,x\in A_i\}.$$

Relation to Complement

For $A,B\subseteq U$, the complement of the intersection is the union of the complements:

$$(A\cap B{)}^c=A^c\cup B^c.$$

Equivalently,

$$A\cap B=(A^c\cup B^c{)}^c.$$

This is a direct application of complement and De Morgan’s law. First, $A^c$ is everything in $U$ outside $A$. The highlighted crescent is $B\setminus A$, which lies inside $A^c$:

Likewise, $B^c$ is everything in $U$ outside $B$. The highlighted crescent is $A\setminus B$, which lies inside $B^c$:

Now place $A$ and $B$ in the same universe. The highlighted region is $A\cap B$. It is exactly the part of $U$ that is in neither $A^c$ nor $B^c$. Therefore everything else lies in $A^c\cup B^c$, so $(A\cap B{)}^c=A^c\cup B^c$.