Venn diagram

Here you will learn about Venn diagrams, including what they are, what they represent, their uses and key set notation.

Students will first learn about Venn diagrams as part of statistics and probability in high school.

What is a Venn diagram?

A Venn diagram is a data visualization of two or more sets.

For example,

Venn Diagram 1 US

Venn Diagram 2 US

Above are examples of a two set and a three set Venn diagram with the following features:

  • The universal set is a rectangle outlining the space in which all values within the smaller sets are found. The universal set is denoted using the symbol \xi.
  • The set \textbf{A} , shown using a circle and labeled A.
  • The set \textbf{B} , shown using a circle and labeled B.
  • The set \textbf{C} , shown using a circle and labeled C.
  • Set A and set B ( and set C) overlap, showing the items which are in set A and in set B. This is called the intersection.

To analyze data using Venn diagrams, all of the values within each set must be correctly placed into the correct part of the Venn diagram. The number of sets is usually outlined or deduced from the information provided.

[FREE] Venn Diagrams Worksheet

[FREE] Venn Diagrams Worksheet

[FREE] Venn Diagrams Worksheet

Use this quiz to check your students’ understanding of using Venn diagrams. 10+ questions with answers covering Venn diagrams topics to identify areas of strength and support!

DOWNLOAD FREE
x
[FREE] Venn Diagrams Worksheet

[FREE] Venn Diagrams Worksheet

[FREE] Venn Diagrams Worksheet

Use this quiz to check your students’ understanding of using Venn diagrams. 10+ questions with answers covering Venn diagrams topics to identify areas of strength and support!

DOWNLOAD FREE

How to use a Venn diagram

Constructing a Venn diagram

To construct a Venn diagram, you draw a region containing two or more overlapping circles (or ellipses), each representing a set. Then fill in the relevant information into each part of the Venn diagram.

For example, this Venn diagram shows the set of numbers \xi={1, 2, 3, 4, 5, 6, 7, 8, 9, 10} which have been sorted into factors of 10 (F) and even numbers (E):

Venn Diagram 3 US

  • All numbers 1-10 are represented on the Venn diagram
  • The factors of 10 appear within F
  • The even numbers appear within E
  • The numbers that are both factors of 10 and even numbers appear in the intersection of F and E
  • The numbers that are not factors of 10 or even numbers are outside of the circles

Set notation

To describe a subset, you need to understand key symbols and set notation for different sets including the intersection of sets, the union of sets and the absolute complement of sets.

Here are some common examples of set notation and their meaning:

Venn Diagram 4 US

You can use combinations of these, for example

A^{\prime} \cup B^{\prime} or (A \cap B)^{\prime}

Subsets

To calculate the number of items in a subset of a Venn diagram, add together the frequencies of the required subset.

For example, this Venn diagram shows the number of people who own a cat (C) or a dog (D).

Venn Diagram 5 US

To find the total number of people who own a dog, you need to include everyone in set D. The number of people who own a dog is 12+6=18.

Venn Diagram 6 US

To find the total number of people who own a dog or a cat, you need to include everyone in set C and set D. The number of people who own a dog or a cat is 12+6+9=27.

Venn Diagram 7 US

Calculating probabilities

To calculate a Venn diagram probability, you need to know each frequency within a subset and the frequency of the larger set which the probability is conditioned on.

For example, this Venn diagram shows the number of people who like Chinese food (C) and Indian food (I):

Venn Diagram 8 US

Find the probability that a person likes Chinese and Indian food.

In general, probability =\cfrac{\text{number of desired outcomes}}{\text{total number of outcomes}}

Here, the probability that a person chosen at random likes Chinese and Indian food, written P(C\cap{I}), is \cfrac{34}{80} since 34 people like Chinese and Indian food out of 80 people in total.

What is a Venn diagram?

What is a Venn diagram?

Common Core State Standards

How does this relate to high school math?

  • Statistics and Probability – Conditional Probability and the rules of Probability (HS.S-CP.A.1)
    Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

  • Statistics and Probability – Conditional Probability and the rules of Probability (HS.S-CP.A.2)
    Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

  • Statistics and Probability – Conditional Probability and the rules of Probability (HS.S-CP.A.3)
    Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

How to use Venn diagrams to solve problems

Each example below provides a step-by-step guide on how to solve specific types of problems involving Venn diagrams.

Venn diagram examples

Example 1: constructing a two set Venn diagram

The set \xi={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.

Draw a Venn diagram that shows the distribution of the values in the set \xi into the two following sets:

  • Prime Numbers
  • Factors of 12

  1. Draw the rectangular universe.

Venn Diagram 9 US

2Draw the overlapping circles within the universe and label.

The first circle should be labeled with the name of the set Prime Numbers. The second circle should be labeled Factors of 12.

Venn Diagram 10 US

3Fill in the information for each subset within the Venn diagram.

Place each item in the universal set into the correct part of the Venn diagram.

Top tip: Go through each item in order so you don’t miss any values.

  • 1 is a factor of 12 and not a prime number. It goes in the right crescent of the set Factors of 12.
  • 2 is a factor of 12 and it is a prime number. It goes in the overlapping part of the two circles (the intersection).
  • 3 is a factor of 12 and it is a prime number. It goes in the intersection of the two sets.
  • 4 is a factor of 12 and not a prime number. It goes in the right crescent.
  • 5 is a prime number but not a factor of 12. It goes in the left crescent.

So far you have the following Venn diagram:

Venn Diagram 11 US

Continuing this process, you get the final solution:

Venn Diagram 12 US

Example 2: constructing a three set diagram

A research group divided 40 construction workers into three groups based on the following criteria:

  • A= {Aged over 30 years old}
  • B= {Builder}
  • C= {Owns a car}

10\% of the participants satisfy all three criteria.

\cfrac{1}{2} of the construction workers satisfy at least 2 of the three criteria.

Complete the following three circle Venn diagram to represent the distribution of the participants in the research study.

Venn Diagram 13 US

Draw the rectangular universe.

Draw the overlapping circles within the universe and label.

Fill in the information for each subset within the Venn diagram.

Example 3: describing a subset of a two set Venn diagram

State the name of the subset that is the shaded region in the Venn diagram below:

Venn Diagram 17 US

Determine which sets are included in the shaded region.

Determine which sets are not included in the shaded region.

State the set notation for the shaded region.

Example 4: set notation

A meteorologist recorded the daily weather patterns over the month of November. The two factors that he wants to look at are:

  • S= {number of days with at least 4 hours of sunshine}
  • T= {Temperature dropped below 32^{\circ}F between 8 pm and 8 am}

Below is a Venn diagram of her results.

Determine the frequency of the set S\cup{T}^{\prime}.

Venn Diagram 18 US

Calculate the frequencies in each subset of the Venn diagram.

Add all of the frequencies within the two sets.

Example 5: probability

40 musicians were asked if they could play the Violin (V) or the Piano (P). The results are shown in the Venn diagram below.

Venn Diagram 20 US

Calculate the probability of selecting a musician at random that can only play the violin.

Determine the parts of the Venn diagram that are in the subset.

Calculate the frequency of the subset.

Calculate the total frequency of the larger set.

Write the probability as a fraction, and simplify.

Example 6: conditional probability

70 students were asked about whether they play B= {basketball} or S= {soccer}. The results are shown in the Venn diagram below.

Venn Diagram 22 US

Calculate the probability of a student not playing soccer given that they play basketball.

Determine the parts of the Venn diagram that are in the subset.

Calculate the frequency of the subset.

Calculate the total frequency of the larger set.

Write the probability as a fraction and simplify.

Teaching tips for Venn diagram

  • After introducing Venn diagrams, let students brainstorm different ways they can be used in mathematics or the real world.

  • Incorporate activities that encourage collaboration and allow students to listen to and critique the ideas of others.

  • Provide Venn diagram templates for students to fill in or provide them with digital diagram makers, instead of asking students to draw their own circles. This can prevent issues for students who draw circles too small or too big.

Easy mistakes to make

  • Placing every item in a set in the intersection
    The intersection is the overlap between two sets. If the item is only in Set A and not Set B, it must go in the subset that is just A, and not the intersection.

    Venn Diagram 24 US
    ( This is the intersection of sets A and B'. We write this as A\cap{B}^{\prime} and say A and not B, or just A)

  • Missing values
    Make sure that every value within all of the sets is written into the Venn diagram. Count the items or add the frequencies to check your answer.

  • Labeling sets incorrectly
    Take for example the following two sets:
    • Odd numbers less than 10
    • Prime numbers less than 10
      The Venn diagram is drawn so that the two sets are given as:
      Venn Diagram 25 US
      but the Venn diagram should look like this:
      Venn Diagram 26 US

  • Counting the intersection values twice
    The items that are placed in the intersection are sometimes incorrectly added twice, when students incorrectly assume that the values are doubled since they belong to two sets. They only need to be counted once.

  • Using probability terminology incorrectly
    The set of A is written as A which consists of a list of items or a frequency. The probability of A is written as P(A) and is a fraction. These two should not be confused.

  • Choosing the universal set, instead of the subset for conditional probability
    For conditional probability, the denominator of the fraction will be a subset of the population, rather than the universal set.

Practice Venn diagram questions

1. Let x be an integer. Construct a two set Venn diagram to represent the set \xi=\{0<x\leq{20}\} shared between the set of even numbers and the set of the multiples of 3.

Venn Diagram 27 US

GCSE Quiz False

Venn Diagram 28 US

GCSE Quiz False

Venn Diagram 29 US

GCSE Quiz False

Venn Diagram 30 US

GCSE Quiz True
  • The even numbers that are not multiples of 3 are 2, 4, 8, 10, 14, 16, 20. These are in the left crescent representing only even.
  • The multiples of 3 that are not even numbers are 3, 9, 15. These are in the right crescent representing only multiples of 3.
  • The even numbers that are also multiples of 3 are 6, 12, and 18. These are where the circles overlap.
  • The numbers that are not even or multiples of 3 are: 1, 5, 7, 11, 13, 17, and 19.

 

Venn Diagram 31 US

2. A group of people were surveyed about their phobias.

 

  • 30 people had a phobia of balloons (B).
  • 24 people had a phobia of spiders (S).
  • 15 people had a phobia of clowns (C).
  • 5 people had a phobia of balloons, spiders and clowns.

 

Which Venn diagram shows the correct results of the survey?

Venn Diagram 32 US

GCSE Quiz False

Venn Diagram 33 US

GCSE Quiz True

Venn Diagram 34 US

GCSE Quiz False

Venn Diagram 35 US

GCSE Quiz False
  • Since 5 people have a phobia of all 3, 5 is placed in the center of the Venn diagram where all of the three circles intersect.
  • The total for balloons is 30 so 30-(4+5+9)=12. ( Just B)
  • The total for clowns is 15 so 15-(0+5+4)=6. ( Just C)
  • The total for spiders is 24 so 24-(0+5+9)=10. ( Just S)

3. Determine the shaded region on the Venn diagram for the set A^{\prime}\cap{B}^{\prime}.

GCSE Quiz False

Venn Diagram 37 US

GCSE Quiz False

Venn Diagram 38 US

GCSE Quiz False

Venn Diagram 39 US

GCSE Quiz True

The set A^{\prime}: Venn Diagram 40 US

 

The set B^{\prime}: Venn Diagram 41 US

 

And so the set A^{\prime}\cap{B}^{\prime} is the intersection of the two sets A^{\prime} and B^{\prime} or the regions that occur in both sets. This is the region:

 

Venn Diagram 42 US

4. A local pharmacy is carrying out some research. They would like to find out how many people have had one of the following three symptoms of flu:

 

  • H= {Headache}
  • C= {Cough}
  • F= {Fever}

 

Venn Diagram 43 US

 

Calculate the frequency of people who had a cough or a fever, but not a headache.

222
GCSE Quiz False

614
GCSE Quiz False

734
GCSE Quiz False

274
GCSE Quiz True

178 people had a cough only. 44 people had a fever only. 52 people had a cough and a fever, but not a headache. The number of people that had a couch or a fever, but not a headache is 178+52+44=274.

5. The Venn diagram below shows the names of 2D shapes that have at least one pair of parallel sides (P) or are quadrilateral (Q).

 

Venn Diagram 44 US

 

From the set \xi, determine P(P\cup{Q})^{\prime}.

\cfrac{9}{11}
GCSE Quiz False

\cfrac{3}{8}
GCSE Quiz False

\cfrac{8}{11}
GCSE Quiz False

\cfrac{3}{11}
GCSE Quiz True

The set (P\cup{Q})^{\prime} contains all of the values that are not in the set P or Q:

 

Venn Diagram 45 US

 

The three shapes that are in the shaded region are: equilateral triangle, regular pentagon, and scalene triangle.

 

As there are 11 shapes in the Venn diagram,

 

P(P\cup{Q})^{\prime}=\cfrac{3}{11}.

6. A university’s entry requirements to study a degree in Medical Science is a grade A in Math, and then a grade A in at least two of the three subjects: Chemistry, Biology, or Physics. The Venn diagram below shows the frequencies of students who applied for the course and achieved an A grade in each subject.

 

Venn Diagram 46 US

 

Calculate P(C given that B\cup{P}).

\cfrac{14}{41}
GCSE Quiz True

\cfrac{20}{41}
GCSE Quiz False

\cfrac{2}{3}
GCSE Quiz False

\cfrac{14}{47}
GCSE Quiz False

The frequencies within each shaded region for the required subsets are:

 

C\cap(B\cup{P}): Venn Diagram 47 US

 

\begin{aligned} &=8+2+4 \\\ &=14 \end{aligned}

 

B\cup{P}: Venn Diagram 48 US

 

\begin{aligned} &=8+11+2+9+4+7 \\\ &=41 \end{aligned}

 

P(C given that B\cup{P})=\cfrac{14}{41}

Venn diagram FAQs

Can there be four or more circles in a Venn diagram?

Yes, Venn diagrams are multi-circle diagrams and there can be as many circles as the context requires, which includes four sets or more.

What are other names for a Venn diagram?

The terms ‘set diagram’ and ‘logic diagram’ are also used to refer to Venn diagrams.

What is set theory?

A branch of mathematics that deals with sets, which are collections of groups or objects.

The next lessons are

A brief history of diagrams

Logician John Venn (1834-1923) was also a mathematician and philosopher who introduced the Venn diagram which is widely used in modern day statistics, probability, set theory and computer science.

Clarence Irving wrote about the Venn diagram in his book ‘A survey of symbolic logic’ in 1918 after which Venn diagrams became more popular.

The Venn diagram below shows the relationship between the data set of Factors of 18, and the data set of Prime numbers less than 18.

Venn Diagram 49 US

Leonhard Euler (1707-1783) was a pioneering Swiss mathematician who has greatly influenced today’s fundamental understanding of mathematics in various fields such as fluid mechanics, complex analysis and calculus.

An Euler diagram (sometimes referred to as Eulerian circles) is different to a Venn diagram because they only show relevant relationships, whereas a Venn diagram shows all possible relationships.

Take for example the set of 2D shapes and 3D shapes. These two sets cannot overlap as a 2D shape cannot be a 3D shape, however quadrilaterals are a subset of 2D shapes and so they are contained entirely within the set of 2D shapes.

Venn Diagram 50 US

A Carroll diagram (named after the author and mathematician Lewis Carroll) is similar to a two-way table which categorizes items within a set into smaller subsets.

For example, below is a Carroll diagram showing the Humans and Animals that did/not attend the tea party in the book ‘Alice in Wonderland’:

Venn Diagram 51 US

Note: The term Venn diagram is named after John Venn and so ‘Venn’ should be capitalized (this is the same for Euler diagrams and Carroll diagrams as they are named after Leonhard Euler, and Lewis Carroll respectively).

Still stuck?

At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.

Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.

One on one math tuition

Find out how we can help your students achieve success with our math tutoring programs.

x

[FREE] Common Core Practice Tests (3rd to 8th Grade)

Prepare for math tests in your state with these 3rd Grade to 8th Grade practice assessments for Common Core and state equivalents.

Get your 6 multiple choice practice tests with detailed answers to support test prep, created by US math teachers for US math teachers!

Download free