## Venn Diagrams

### Unions, intersections, and relative complements

Given two sets A and B, the union of A and B is the set consisting of all objects which are elements of A or of B or of both. It is written as A u B.

The intersection of A and B is the set of all objects which are both in A and in B. It is written as A n B.

Finally, the relative complement of B relative to A, also known as the set theoretic difference of A and B, is the set of all objects that belong to A but not to B. It is written as A \ B. Symbolically, these are respectively:

- A u B := {x : (x \in A) or (x \in B)}
- A n B := {x : (x \in A) and (x \in B)}
- A \ B := {x : (x \in A) and not (x \in B) }

Notice that B doesn't have to be a subset of A for A \ B to make sense.

To illustrate these ideas, let A be the set of left-handed people, and let B be the set of people with blond hair. Then A n B is the set of all left-handed blond-haired people, while A u B is the set of all people who are left-handed or blond-haired or both. A \ B, on the other hand, is the set of all people that are left-handed but not blond-haired, while B \ A is the set of all people that have blond hair but aren't left-handed.

Venn diagrams are illustrations that show these relationships pictorally. The above example can be drawn as the following Venn diagram: