
# 13.3: Venn Diagrams


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 $$A \cup B, A \cap B,$$ and $$A^{c} \cap B$$

$$A \cup B$$ contains all elements in either set.

$$A \cup B$$ contains all elements in either set.

$$A \cap B$$ contains only those elements in both sets - in the overlap of the circles.

## Example 10

Use a Venn diagram to illustrate $$(H \cap P)^{c} \cap W$$

We'll start by identifying everything in the set $$\mathrm{H} \cap P$$

Now, $$(H \cap P)^{c} \cap W$$ 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 $$\mathrm{H}$$ and $$F$$, but are not in set $$W$$. So we could represent this set as $$H \cap F \cap W$$

Try it Now 3

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

$$A \cup B \cap C^{c}$$