4.3: Venn Diagrams

Learning Objectives

• Use a Venn diagram to illustrate the relationship between 2 and 3 sets
• Create an expression relating 2 and 3 sets from a Venn diagram

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 18^th 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 $\PageIndex{1}$

Create Venn diagrams to illustrate $A \cup B, A \cap B,$ and $A' \cap B$.

Solution

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

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

$A' \cap B$ contains the elements of $B$ that are not in $A$.

Example $\PageIndex{2}$

Use a Venn diagram to illustrate $(H \cap F)' \cap W$.

Solution

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

Now, $(H \cap F)' \cap W$ will contain everything not in the set identified above that is also in set $W$.

Example $\PageIndex{3}$

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

Solution

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 $\PageIndex{1}$

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

Answer

$\left( A \cup B \right) \cap C'$