$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 10.3: Venn Diagrams

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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 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. Ac B will contain the elements in set B that are not in set A. ### Example 10

Use a Venn diagram to illustrate (H F)cW

We’ll start by identifying everything in the set H F Now, (H F)cW 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 FWc ### Try it Now 3

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