What Is A Venn Diagram? (5 Key Things You Should Know)


Venn diagrams are used throughout math, science, and other disciplines to express ideas in a visual way.  Once you know what they are and why to use them, you can figure out how to solve them.

So, what is a Venn diagram?  A Venn diagram is a picture that uses shapes (usually circles) to show the relationship between sets. Each set is represented by a circle, and we can see the overlap between sets (if there is an overlap). We use Venn diagrams for set theory, probability, logic, computer science, & other subjects.

Of course, we can also study the complement of sets (and their unions or intersections) using Venn diagrams.

In this article, we’ll talk about Venn diagrams, when to use them, and how to solve them.  We’ll also answer some common questions about Venn diagrams.

Let’s get started.

What Is A Venn Diagram?

A Venn diagram is a picture that shows the relationship between two or more sets (groups of objects).  A Venn diagram shows the overlap between sets (if there is any).

Venn diagram
A Venn diagram can show us the relationship between two or more sets. Here, we can see the sets A (blue) and B (yellow), along with their overlap (AnB) and their union (AuB, which is A, B, and AnB taken together).

A circle is the most common shape for each set in a Venn diagram.  We can draw individual elements (items) of each set in a Venn diagram.

What Is A Venn Diagram Used For? (When To Use Venn Diagrams)

A Venn diagram is used in lots of different ways in math and science, including:

  • Set Theory – for example, showing the relationship between imaginary numbers, real numbers, and complex numbers.
  • Probability – for example, showing the probability that a person belongs to one group, or another (or both groups, or neither group).
  • Logic – for example, showing the overlap between logical statements (Fido is a dog, Fido is a mammal, Fido is an animal, etc.)
  • Computer Science – for example, showing the overlap between items in databases.
probability venn diagram 1
A Venn diagram can also be used for probabilities of events A and B.

A Venn diagram is used to show the union, intersection, and complement of sets in a visual way.

Where Is The Union On A Venn Diagram?

The union on a Venn diagram is the inside of every set taken together:

  • If there are 2 sets A and B in the Venn diagram, then the union is AuB, and it includes everything inside set A, everything inside set B, and everything inside their overlap.
  • If there are 3 sets A, B, and C in the Venn diagram, then the union is AuBuC, and it includes everything inside set A, everything inside set B, everything inside set C, and everything inside any overlap (A & B, A & C, B & C, or A & B & C).
  • and so on for 4, 5, 6, … sets.

You can see the union for a 2-set Venn diagram illustrated below.

Venn diagram - AuB
The union of two sets A and B (blue) is everything that is one set, or the other, or both.

You can see the union for a 3-set Venn diagram illustrated below.

Venn diagram - AuBuC
The union of three sets A, B, and C (blue) is everything that is in at least one of the sets.

Where Is The Intersection On A Venn Diagram?

The intersection on a Venn diagram is the place where all sets in the diagram overlap (if any such place exists):

  • If there are 2 sets A and B in the Venn diagram, then the intersection is AnB, and it includes everything in the overlap between sets A and B.
  • If there are 3 sets A, B, and C in the Venn diagram, then the intersection is AnBnC, and it includes everything in the overlap between sets A, B, and C.
  • and so on for 4, 5, 6, … sets.

You can see the intersection for various Venn diagrams illustrated below.

You can see the intersection for a 2-set Venn diagram illustrated below.

Venn diagram - AnB
The intersection of two sets A and B (blue) is the overlap, or anything that is in both sets simultaneously.

You can see the intersection for a 3-set Venn diagram illustrated below.

Venn diagram - AnBnC
The intersection of three sets A, B and C (blue) is the overlap, or anything that is in all three sets simultaneously.

Where Is the Complement On A Venn Diagram?

The complement on a Venn diagram is the place outside of a given set (that is, everything not in the set).

For a set A, the complement is denoted by AC.  You can see the set A and its complement illustrated below.

Venn diagram complement
The complement of a set A (blue) is anything that is not in the set A.

We can also take the complement of AuB or AnB by using De Morgan’s Laws:

  • (AuB)C = ACnBC
  • (AnB)C = ACuBC

You can see the complement of AuB below:

Venn diagram complement of AuB
The complement of AuB (blue) is anything outside of the set AuB. By De Morgan’s Law, it is the intersection of AC and BC.

You can also see the complement of AnB below:

Venn diagram complement of AnB
The complement of AnB (blue) is anything outside of the set AnB. By De Morgan’s Law, it is the union of AC and BC.

Can A Venn Diagram Not Overlap?

In some cases, the two sets in a Venn diagram will have no items in common.  In that case, there is no overlap between the circles in the diagram.

This suggests that the set AnB (the intersection of A and B) is empty.  In the case of a probability problem, no overlap between sets A and B in a Venn diagram means that P(AnB) = 0.

It also means that we have a simplified formula for P(AuB):

  • P(AuB) = P(A) + P(B) – P(AnB)  [the usual formula for the probability of AuB]
  • P(AuB) = P(A) + P(B) – 0  [since P(AnB) = 0 when there is no overlap between A and B]
  • P(AuB) = P(A) + P(B)  [simplified formula for P(AuB)]

A lack of overlap can happen when there are 2 sets in the Venn diagram, as shown below.

Venn diagram disjoint sets A and B
The sets A (blue) and B (yellow) are disjoint: they do not overlap, and their intersection AnB is empty.

A lack of overlap can also happen when there are 3 sets in the Venn diagram, as shown below.

Venn diagram disjoint sets A B C
The sets A (blue), B (yellow), and C (red) are disjoint: they do not overlap, and their intersection AnBnC is empty. The intersection AnB, AnC, and BnC are also empty.

Can A Venn Diagram Have One Circle?

A Venn diagram can have one circle.  In this case, we can only illustrate a set A and its complement, AC.  In other words, we can decide if an item is in the set A or outside of the set A.

Venn diagram one circle complement
This Venn diagram has just one circle: inside is the set A (yellow), and outside is the complement of A, the set AC.

Can A Venn Diagram Have 3 Circles?

A Venn diagram can have 3 circles.  In this case, there are 3 sets A, B, and C to be graphed.

There may be no overlap at all between the 3 sets, or there may be some overlap, or they may be “nested” in some way (for example, if A contains B, and B contains C).

Venn diagram 3 sets nested
This Venn diagram has 3 circles. The sets are nested so that A contains B and B contains C (and thus, A contains C).

Can A Venn Diagram Have 4 Circles?

A Venn diagram can have 4 circles.  In this case, there are 4 sets A, B, C, and D to be graphed.  It gets a little tricky to draw the diagram at this stage!

There may be no overlap at all between the 4 sets, or there may be some overlap, or they may be “nested” in some way (for example, if A contains B, B contains C, and C contains D).

Venn diagram 4 circles
This Venn diagram has 4 circles. The sets all intersect at the center to give us the set AnBnCnD.

Can Venn Diagrams Be Square?

Venn diagrams can use squares for the sets (or any shape you like).  It is customary to use a circle for each set in the diagram but you can still express the same ideas with squares and other shapes.

Venn diagram 3 squares
You can use squares or other shapes for a Venn diagram, but the concept is the same.

How To Solve A Venn Diagram

To solve a Venn diagram, we fill in every area on the graph with a number (or with the items) or a probability.

Example 1: How To Solve A Venn Diagram With Two Sets

Let’s say we have the Venn diagram below.

Venn diagram 2 circles
This Venn diagram has two circles. We want to solve for x.

The large rectangle contains 100 items in total. There are 30 items outside of the circles, 40 in the yellow part of the left circle, x in the blue part of the right circle, and 10 in the green overlap.

This gives us the equation:

  • 30 + 40 + x + 10 = 100
  • x + 80 = 100
  • x = 20

So, we have x = 20. This means that the left circle has 40 + 10 = 50 items, while the right circle has 20 + 10 = 30 items total.

Example 2: How To Solve A Venn Diagram With Three Sets

Let’s say we have the Venn diagram below.

Venn diagram 3 circles probability
This Venn diagram displays probabilities. The total probability is 1 (assume every event is within at least one circle).

Since the total probability is 1, we can write the equation:

  • x + 0.2 + 0.15 + 0.15 + 0.1 + 0.2 + 0.15 = 1
  • x + 0.95 = 1
  • x = 0.05

So, x = 0.05, and the probability of an overlap between all three circles is 5% (0.05 as a decimal).

Conclusion

Now you know what a Venn diagram is and what they are used for.  You also know the answers to some common questions about Venn diagrams and how to solve them.

I hope you found this article helpful.  If so, please share it with someone who can use the information.

Don’t forget to subscribe to my YouTube channel & get updates on new math videos!

~Jonathon

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