Venn Diagram

Venn Diagram

Author: Sophia Tutorial

This lesson will explain Venn diagrams.

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What's Covered

This tutorial will cover Venn diagram. You will learn about:

  1. Venn diagrams
  2. Complements
  3. Not Overlapping

1. Venn Diagrams

Venn diagrams are an important idea in probability theory.

Term to Know

Venn Diagram

A way to compare and contrast events from a chance experiment. Two events may both occur, only one or the other may occur, or neither may occur.

Venn diagrams were invented by the logician John Venn. They're useful for understanding the relationships between two different sets of objects or two different events in probability.

A typical Venn diagram with two sets, intersecting sets in it, has four parts to it.

There's the part of it that only is in event A but not in event B. There's a part of it that's only in event B, but not event A. There's the part that's both events A and B. And then there's the part that's neither A nor B.

Think About It

Suppose that in a high school of 80 students, 18 of them take statistics, 15 take economics, and 8 students take both.

So what would that look like in terms of the number of students that go in each of these four areas?

The first instinct of most people is to say 18 goes in the statistics bubble, 15 goes in the economics bubble, eight goes in the both bubble, and whatever's left of the 80 goes on the outside. That graph would look like this:

That's not a horrible idea, but it has a few problems:

  • Start with the innermost portion of the circle, the 8 students that take both statistics and economics.
  • Currently, the diagram indicates that there are 15 students who take economics. But 8 of those students are also taking statistics and they fall into the middle category.
  • Therefore, of the 15 total students taking economics, 8 of them should be in the center category and the remaining 7 are should be in the economics only category.

Apply the same logic to the other side: of the 18 students taking statistics, 8 are also taking econ, which means there are only 10 students taking statistics only.

As you can see, of the 80 students that remain, 55 of them are taking neither statistics nor economics.

2. Complements

We can also show complements in a Venn diagram. Complements are everything that's not in a particular event.

The complement of A is everything outside of the A bubble, including the part of B that does not overlap with A. The overlapping portion is in A. It is therefore not part of the complement. The part of B that's B only and the part that's neither A nor B are the complement of A.

3. Not Overlapping

Not all Venn diagrams need overlap like this. Sometimes, the two events, A and B, don't have any outcomes in common. That looks like this:

Example  Suppose A is the event of rolling an even number and B is the event rolling a five. There's no overlap between the two ideas. You can draw the bubbles as separate. There can be nothing in both. In the neither category would be any rolls that are odd but not five, which leaves one and three.


Many relationships between events can be represented using Venn diagrams: complements, overlap, and not overlapping. Venn diagrams are also useful for probability. Two event Venn diagrams, such as the ones in this tutorial, consist of four areas: the A-only area, the B-only area, the both A and B area, and the neither A nor B area. The total number of items can be found by adding up all the subcategories within that event.

Thank you and good luck!


Terms to Know
Venn Diagram

A way to compare and contrast events from a chance experiment. Two events may both occur, only one or the other may occur, or neither may occur.