Intersection (Set)

set-theory

Definition

Intersection

Let be a universe and let . The intersection of and is the set

More generally, for a family of subsets of ,

Relation to Complement

For , the complement of the intersection is the union of the complements:

Equivalently,

This is a direct application of complement and De Morgan’s law. First, is everything in outside . The highlighted crescent is , which lies inside :

Likewise, is everything in outside . The highlighted crescent is , which lies inside :

Now place and in the same universe. The highlighted region is . It is exactly the part of that is in neither nor . Therefore everything else lies in , so .