To visualize the interaction of sets, John Venn in 1880 thought to use overlapping circles, building on a similar idea used by Leonhard Euler in the 18th century. These illustrations now called **Venn Diagrams**.

### Venn Diagram

A Venn diagram represents each set by a circle, usually drawn inside of a containing box representing the universal set. Overlapping areas indicate elements common to both sets.

Basic Venn diagrams can illustrate the interaction of two or three sets.

### Example 9

Create Venn diagrams to illustrate *A *⋃* B*, *A *⋂* B*, and *Ac *⋂* B*

*A *⋃* B* contains all elements in *either* set.

*A *⋂* B* contains only those elements in both sets – in the overlap of the circles.

*Ac *will contain all elements *not* in the set A. *A ^{c }*⋂

*B*will contain the elements in set

*B*that are not in set

*A*.

### Example 10

Use a Venn diagram to illustrate (*H *⋂* F*)^{c} ⋂* W*

We’ll start by identifying everything in the set *H *⋂* F*

Now, (*H *⋂* F*)*c* ⋂* W* will contain everything *not* in the set identified above that is also in set *W*.

### Example 11

Create an expression to represent the outlined part of the Venn diagram shown.

The elements in the outlined set *are* in sets *H* and *F*, but are not in set *W*. So we could represent this set as *H *⋂* F* ⋂* W ^{c
}*

### Try it Now 3

Create an expression to represent the outlined portion of the Venn diagram shown