7.3: Venn Diagrams

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Mar 3, 2021
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- 7.2: Union, Intersection, and Complement
- 7.4: Cardinality

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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 9

Create Venn diagrams to illustrate $A \cup B, A \cap B,$ and $A^{c} \cap B$

$A Venn diagram showing two overlapping sets A and B. The region included in either set is highlighted.$ $A \cup B$ contains all elements in either set.

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

$A Venn diagram showing two overlapping sets A and B. The overlapping region included in both sets is highlighted.$

$A Venn diagram showing two overlapping sets A and B. The region included only in B but not A is highlighted.$ $A \cap B$ 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 $\mathrm{H} \cap P$

$A Venn diagram of three sets H F and W are shown overlapping. The region where H and F overlap is highlighted.$

Now, $(H \cap P)^{c} \cap W$ will contain everything not in the set identified above that is also in set $W$

$A Venn diagram of three sets H F and W are shown overlapping. The region in set W is highlighted, with the exception of the part that also lies in both F and H. In other words, all of W, except for the part where all three overlap.$

Example 11

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

$A Venn diagram of three sets H F and W are shown overlapping. The region where H and F overlap, but W does not, is highlighted.$ 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

$A Venn diagram of three sets A B and C are shown overlapping. The region highlighted includes anything in A or anything in B, excluding anything also in C$

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

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Venn Diagram

Example 9

Example 10

Example 11

Try it Now 3

Venn Diagram

Example 9

Example 10

Example 11

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