In this lesson, you will learn what Venn diagrams are in math. You will also learn the history behind them and will be given the different types illustrated with examples. Following this lesson will be a brief quiz to test your knowledge.

## What Is a Venn Diagram?

English logician, John Venn, was the inventor of the Venn diagram in 1880.

He constructed the Venn diagram to help illustrate inclusion and exclusion relationships between sets, except he did not call it the ‘Venn diagram.’ He called the circles ‘Eulerian circles.’ Clarence Lewis referred to the diagram as the Venn diagram in his book, *A Survey of Symbolic Logic* in 1918.**Venn diagrams** are illustrations of circles that depict commonalities or differences between two sets. What are sets, you may ask? A **set** is simply a grouping or collection of items. The items in a set are actually called **elements**. You indicate elements in a set by putting {brackets} around them.

For a quick example, if you have the set {Andrew, Tyler, Michelle} of people in your music class and the set {Leo, Ava, Lia} of people in your science class, you can put these sets in circles to better illustrate who is in each class.

You can clearly see who is in your music and science class and who is in your science class. But, if you wanted to show that Tyler and Leo were in both your music and science class, you could use a Venn diagram.

It’s important to note what a universal set is. If there was a rectangle outside of the Venn diagram, that encompassed both circles of sets, it would be called the **universal set**. The universal set is indicated by the capital *U* in the image. In our example, we could say that the universal set is ‘school.’

A Venn diagram is not always two circles. For more complicated problems and situations, Venn diagrams could be several circles.

## Types of Venn Diagrams

Venn diagrams are helpful in illustrating several types of relationships.

#### Disjoint sets

Taking the example of the science and math class from before, the initial diagram represents disjoint sets because the two sets (science and music class) have no commonalities. The students in music class are only in your music class, and the students in science class are only in your science class. There is no relationship between the two. Disjoint sets are always represented by separate circles.

#### Intersections

Taking the disjoint set Venn diagram example from before, if you overlap the circles, it indicates an intersection.

The intersection represents elements that are in both sets. It represents the commonality between sets. So, in the music and science class example, the intersection is indicated in gray and means that Tyler and Leo are in both your music and science classes.

An intersection forms another set.

It can be indicated in writing by an upside down *U* and will look like this:

If we went back to the example of the two disjoint sets with two separate circles, they would form no intersection. This is called the empty set and can be written like this:

The circle with the slanted line through it means that there is an empty set due to disjoint sets.

#### Unions

A union is the set of elements that are contained in both sets, only included once. Let’s look at the Venn diagram with overlapping sets *A*, *B* and *C*.

We would write this union by including each element of *A*, *B* and *C* only once, even if there are repeat elements within the sets. It would look like this:

#### Complements

Complements are ways of saying ‘everything that is not.’ It is indicated by a little *c*. Using the Venn diagram from the unions section, if we wanted to show ‘everything that is not in *A*,’ we could shade every other area except for *A*.

This Venn diagram representing *A* complement can be written like this:

#### Subsets

Venn diagrams can be used to indicate subsets.

If a small circle is within a bigger circle, it can be said that the smaller circle encompasses the properties of the bigger circle and is thus a subset of the bigger circle. In this illustration, the large circle encompasses all animals with wings. The smaller circle is dinosaurs with wings, which is a subset of animals with wings. The larger circle is actually a subset of the universal set, which is ‘animals.’

## Lesson Summary

**Venn diagrams** were created by John Venn in 1880, though they were not called Venn diagrams until Clarence Lewis called them that in his book, published in 1918. They are circles that show commonalities and differences between two or more sets.

A **set** is a group of items, indicated by {brackets}. There are five times of Venn diagrams. A diagram is called **disjoint** if there are no commonalities, meaning there are two separate circles. **Intersections** are where circles overlap to show a commonality. **Unions** are sets of elements contained in both sets, but only included once, meaning you write them out only once, even if they appear multiple times.

**Complements** show everything that is not a specific set. Finally, **subsets** show a set that is part of a larger set.