A Venn diagram is one way we have of displaying sets. The advantage of a Venn diagram is that it can show logical relationships between different sets. For our use, we will mainly use Venn diagrams to display logic, as well as probabilities.

In a Probability setting, the set of everything we are considering in a Venn diagram is called the sample space, and we denote the sample space by 𝛏 (the lowercase Greek letter xi). An example of a set would be students in a school or people going to the gym. More generally, we take a set to be a group of data points in Statistics.

We often only need to consider two types of Venn diagrams; those with two subsets and those with three.

How do you draw a Venn diagram?

To draw a Venn diagram, we can follow a series of logical steps. We will first look at drawing a diagram for two subsets of the sample space. For now, we will call these A and B.

Step 1: Draw two circles, which have an overlapping section in the middle.

Step 2: Draw a rectangle around the outside of these two circles, leaving room between the circles and the rectangle.

Step 3: Label the rectangle with 𝛏, do this on the outside. This rectangle represents our whole sample space.

Step 4: Label one circle A, and the other B, as these represent our two subsets.

The result should look like this:

Example Venn diagram with two subsets of the sample space, Tom Maloy - StudySmarter Originals

For three subsets of the sample space, say A, B, and C, we follow similar steps but with three circles overlapping, as seen below.

Example Venn diagram with three subsets of the sample space, Tom Maloy - StudySmarter Originals

Examples of Venn diagrams

To get a grasp on how Venn diagrams work, let’s have a look at the following Venn diagram examples.

Logic operators on sets

When we have multiple sets acting together, we can describe parts of them using logic operators. All of these symbols are common whenever we discuss Venn diagram Notation.

Union of sets

The union of sets, denoted by ∪, is a logic operator we can use on sets. When we see this operator, we should think of replacing it with ‘or’. For example, if I denoted A=people who like rugby and B= people who like cricket, and I wrote the statement A∪B, we would interpret this as ‘the people who like cricket or the people who like rugby or the people who like both’. On a Venn diagram, if I were to shade A∪B, it would look like this:

Union of 2 sets, Tom Maloy, Study Smarter Originals

So for an object to be in A∪B, it needs to be in either set.

Intersection of sets

The intersection of sets is a different logic operator we use on sets and is denoted by ∩. When we see this operator, we should think of replacing it with ‘and’. With A and B denoted as above, if we saw A∩B, we should interpret this as ‘People who like cricket and rugby’. On a Venn diagram, it would look like this:

Intersection of 2 sets, Tom Maloy, Study Smarter Original

So for an object to be in A∩B, it needs to be in both sets.

Complement of sets

The final logic operator we need to know about is the complement, which we should think of as ‘not’. If I wanted to write the complement of A, I could write either A’ or $\overline{A}$.

Again with A as above, we would interpret $\overline{A}$ as ‘the set of people who don’t like rugby’. On a Venn diagram, this would look like this:

Complement of a set, Tom Maloy, Study Smarter Originals

How is probability represented using Venn diagrams?

As we mentioned earlier, Venn diagrams can help us determine probabilities. For example, if we look at example 2 from above, we could ask a question such as ‘find the Probability of a student only studying one of maths, physics or chemistry.’ In this case, we would use the general probability equation:$P(eventhappening)=\frac{timeseventhappens}{totalnumberofevents}$, which gives us $\frac{11+14+6}{11+14+6+1+2+8+5+3}=\frac{31}{50}$.

Two events are mutually exclusive if they cannot happen at the same time. In a Venn diagram, this would mean the intersection is empty, and if A and B are mutually exclusive, P(AB)=0.

Logic relations on a Venn diagram

We can also be asked to shade a Venn diagram based on a logic relation, and we have a method to do this, as shown below.

Suppose we had a logic relation between two sets, say A and B. (e.g. A’∪B or A∩B’)

Step 1: Shade the logic relation on A in one colour.

Step 2: Shade the logic relation on B in a different colour.

Step 3: If the two events (A and B) are related by a union, our diagram showing the relation would be wherever there is any shading. If it is an intersection, it would be where there is shading of both.