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,

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.

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):

• 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:

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).

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.

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):

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?

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.

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.

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:

Continuing this process, you get the final solution:

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.

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

Since 10\% of participants are within all three sets, the very center of the Venn diagram where all three sets overlap would contain the value:

(40\div{100})\times{10}=4

Since \cfrac{1}{2} of the participants are in at least 2 of the three sets, 40\times\cfrac{1}{2}=20. The missing value in the intersection between A and C is therefore 20-(2+4+5)=9.

The final missing value for set B is the number of remaining participants. This is equal to 40-(8+3+2+9+4+5+2)=7.

This gives us the completed 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:

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}.

Calculate the frequencies in each subset of the Venn diagram.

The subset S\cup{T}^{\prime} is the union of all of the values in the set S or all of the values not in the set T. This is the following shaded region:

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.

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.

The subset of musicians that can only play the violin is the set:

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.

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.

Identify the subset which is the part of B which is not a part of S:

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.
  
  

  

  ( 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:
    

    

    but the Venn diagram should look like this:
    

    

• 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.
  

Related probability lessons

• Probability notation
• How to find probability
• How to calculate probability
• Probability formula
• Independent events
• Mutually exclusive events
• Probability scale
• Tree diagram probability
• Conditional probability

The next lessons are

• Compound events
• Probability distribution
• Units of measurement

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.

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.

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’:

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).

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