What Are Venn Diagrams in Math? - History, Types & Examples

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  • 0:00 What Is a Venn Diagram?
  • 1:44 Types of Venn Diagrams
  • 4:17 Venn Diagrams and Word…
  • 5:44 Lesson Summary
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Lesson Transcript
Instructor: Karin Gonzalez

Karin has taught middle and high school Health and has a master's degree in social work.

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.

Two circles of music and science classmates.

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.

You can see that Tyler and Leo are in both your music and science class.
Music and Science 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.'

You can see the universal set indicated by a capital U.
Universal set.

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.

Disjoint sets.

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.

The intersection is indicated by the area shaded in gray.
Intersection in a Venn diagram.

An intersection forms another set. It can be indicated in writing by an upside down U and will look like this:

The shaded gray area including Tyler and Leo can be written like this with an upside-down U indicating the intersection of both sets.
The shaded gray intersection including Tyler and Leo can be written like this with an upside-down U indicating the intersection.

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:

Empty Set.

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.

Venn diagram with overlapping sets A, B and C.
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:

Each element is included only once when writing a union.
How the Union is written.

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 represents everything that is not in A by shaded gray.
Complement.

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

Complement of A written.

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