4.3: Venn Diagrams

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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.

$clipboard_ee03de8fb983d8e26d6f038c378cb00be.png$

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

$clipboard_e851dd1ad67dd86f3e5fe2710c0936089.png$

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

$4.3Venn.png$

Example $\PageIndex{2}$

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

Solution

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

$clipboard_ebe59da114cdbdec91c71a3e8167fd89c.png$

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

$clipboard_e5bcfbdfa49842a14d119067dd23ef8cb.png$

Example $\PageIndex{3}$

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

$clipboard_e7f0f9dcccbf62d610ee286fa70153e21.png$

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

$clipboard_ee6f4b6ece2cde3e7320d09311c24e49b.png$

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

Example $\PageIndex{4}$

Let $\quad A=\{ 1, 2, 3 \} \quad B=\{3, 4, 5, 6\} $ with the universal set $ U=\{1, 2, 3, 4, 5, 6, 7\} $

Draw a Venn diagram for the sets A and B in U
What is in the intersection of A and B?
What is not in the union of A and B?

Solution

$Venn_example5.png$

The intersection of A and B contains the number 3.

The number 7 is not in the union of A and B.

Learning Objectives

Venn Diagram

Example \(\PageIndex{1}\)

Solution

Example \(\PageIndex{2}\)

Solution

Example \(\PageIndex{3}\)

Solution

Try It \(\PageIndex{1}\)

Example \(\PageIndex{4}\)

Solution