Venn Diagram

Do you know what a Venn diagram looks like? Do you know who invented them?

We will learn all about them in this lesson.

Venn diagrams were invented by John Venn. He used them to show the similarities and differences between various sets visually.

Venn diagram representing night and day

In this mini-lesson, we will explore the world of the Venn diagram by finding answers to the questions like what are Venn diagram symbols, Venn diagram application in math, and Venn diagram examples while discovering interesting facts around them.

Lesson Plan

What Is a Venn Diagram?

A Venn diagram is a diagram that helps us visualize the logical relationship between sets and their elements and helps us solve examples based on these sets.

A Venn diagram typically uses intersecting and non-intersecting circles (although other closed figures like squares may be used) to denote the relationship between sets.

Let us observe a Venn diagram example.

Here is the Venn diagram of the Universe.

Venn diagram example representing the universe, solar system, Earth, and USA

The correlation between the following set of numbers is easily understood using a Venn diagram.

Venn diagram example of even numbers and numbers in the 5 table

Subset

Let's look at another example. 

There are two sets \(A\) and \(B\) in the figure below if \(A\) and \(B\) are two sets. 

\(A\) represents all the students in grade 5 while \(B\) represents students of grade 5 who can swim.

If every element of set \(B\) belongs to set \(A\), then \(B\) will be called a subset of \(A\).

Subset in Venn diagram - students in grade 5 and those who can swim

This relationship is symbolically represented as \(\text A \subseteq \text B\). 

It is read as "\(B\) is a subset of \(A\)" or "\(B\) subset \(A\)."

Every set is a subset of itself. i.e. \(A \subseteq A\). 

Here is another example of subsets. 

\(N\) = set of natural numbers

\(I\) = set of integers

Here N \(\subseteq \) I, because all natural numbers are integers. 


What Are Venn Diagram Symbols?

There is a long list of about 30 Venn diagram symbols.

For an initial understanding, we shall stick to the below-given basic symbols.

1. The union symbol - \(\cup\)

2. The intersection symbol -  \(\cap\)

3. The complement symbol - \(A\)c or \(A\)'

Venn diagram symbols for union, intersect, complement

Let us now work on the data presented and learn the use of the symbols with this Venn diagram.

Venn diagram example - Students who like burgers, hotdogs, and pizzas

Symbol It refers to Total elements (No. of students)

\(A\) \(\cup\) \(C\)

The number of students that prefer both burger or pizza and not a hotdog. 1 + 2 + 9 = 12
\(A\) \(\cap\) \(C\) The number of students that prefer both burger and pizza and not a hotdog. 2
\(A\) \(\cap\) \(B\) \(\cap\) \(C\) The number of students that prefer a burger, pizza as well as hotdog. 2
\(A\)c or \(A\)' The number of students that do not prefer a burger. 10 + 6 + 9 = 25

How to Draw a Venn Diagram?

To draw a Venn diagram, first, the universal set should be known. 

The universal set is usually represented by a rectangle.

Every set is the subset of the universal set (\(U\)).

Thus, every other set will be placed inside the rectangle.

The other subsets are represented by closed figures usually circles.

Universal set

Think of a bigger set that will accommodate all the given sets under consideration which in general is known as the Universal set.

Generally, the universal set is denoted by \(U\), \(E\), or \(\xi\) and in the Venn diagram, it is represented by a rectangle. 

Universal set - set of students who play soccer

Examples

All the elements of set \(A\) are inside the circle. Set \(A\) is a subset of the universal set \(U\).

They are part of the rectangle which makes them the elements of set \(U\).

Thus, set \(A\) will be represented as follows:

\(A\) \(\cup\) \(U\) = \(U\)

How to draw a Venn diagram

From the above diagram, it is also clear that \(A\) + \(A\)' = \(U\)

What will be the complement of a complement set? It will be set \(A\) itself.

Therefore (\(A\)')' = \(A\)

No elements are common between a set and its complement.

 
Challenge your math skills
Challenging Questions
  1. \(U\) = {a, b, c}, \(A\) = {d, e, f, g}, \(B\)  = {b, d, f}, and \(C\) = {c, e, f, g, i}. Populate a Venn diagram and use this information to find [(\(A\) \(\cup\) \(C\)) \(\cup\) \(B\)].
  2. In a class of 80 students, 65 students like to play basketball. 15 students like to play chess. All the students like to play at least one of the two games. How many students like to play basketball or chess? How many students like to play only basketball?

What Are the Advantages of Using Venn Diagrams?

There are several advantages to using Venn diagrams. 

  • We can visually organize information to see the relationship between sets of items, such as commonalities and differences, and to depict the relations for visual communication. 

Venn diagram of cold and warm blooded aquatic animals

  • We can compare two or more subjects and clearly see what they have in common versus what makes them different. This might be done for selecting an important product or service to buy.
  • Mathematicians also use Venn diagrams in math to solve complex equations.
  • We can use Venn diagrams to compare data sets and to find correlations.
  • Venn diagrams can be used to reason through the logic behind statements or equations.
 
important notes to remember
Important Notes
  1. Every set is a subset of itself i.e., \(A\) \(\subseteq \) \(A\).
  2. \(A\) universal set accommodates all the sets under consideration.
  3. If \(A\) \(\subseteq\) \(B\) and \(B\) \(\subseteq \) \(A\), then \(A\) = \(B\)
  4. The complement of a complement is the given set itself.

Solved Examples

Example 1

 

 

\(A\) set representing various types of fruit is given as follows: {guava, orange, mango, custard apple, papaya, watermelon, cherry}

Venn diagram solved example - basket of fruit

Show these subsets.

  1. Fruit with one seed
  2. Fruit with more than one seed

Solution

Among the various types of fruit, only mango and cherry have one seed.

Thus, 

  1. Fruit with one seed = {mango, cherry}
  2. Fruit with more than one seed = {guava, orange, custard apple, papaya, watermelon}
Example 2

 

 

\(A\) = {1, 3, 5, 7}, \(B\) = {0, 1, 2, 4, 5, 6, 8} are two subsets of the universal set \(U\) = {0, 1, 2, 3, 4, 5, 6, 7, 8}.

Julia drew a Venn diagram representing the following relations but she is confused about the following operations. Can you help her?

  1.  \(A\) \( \cup\) \(B\)
  2.  \(A\) \( \cap\) \(B\)
  3.  \(A\) \( \cup\) \(B\)'

Solution

The Venn diagram for the above relations can be drawn as:

Venn diagram solved examples

  1.  \(A\) \( \cup\) \(B\) = {0, 1, 2, 3, 4, 5, 6, 7, 8} 
  2.  \(A\) \( \cap\) \(B\) = {1, 5}
  3.  \(A\) \( \cup\) \(B\)' = {1, 3, 5, 7} \( \cup\) {3, 7} = {1, 3, 5, 7}

Note: \(B\)' = Compliment of \(B\)

There are no elements common between a set and its complement.

 \(\therefore\) \(A\) \( \cup\) \(B\) = {1, 2, 3, 4, 5, 6, 7, 8}

 \(A\) \( \cap\) \(B\) = {1, 5}

 \(A\) \( \cup\) \(B\)' = {1, 3, 5, 7}

Example 3

 

 

\(P\) is a set of prime numbers 3, 5, 7, 11, and 13, and \(F\) is a set of factors of 27.

Help Nathan list the elements based on the given Venn diagram.

P intersection F

List the elements of:

  1. \(U\)
  2. \(P\)'
  3. \(F\)'
  4. \(P' \cup F\)
  5. \(P \cap F\)

Solution

\(P\) = {3, 5, 7, 11, 13}

\(F\) = {1, 3, 9, 27}

P intersection F

  1. \(U\) = \( P \cup F \) = {1, 3, 5, 7, 9, 11, 13, 27}
  2. \(P\)' = {1, 9, 27}
  3. \(F\)' = {5, 7, 11, 13}
  4. \(P' \cup F\) = {1, 9, 27} \( \cup\) {1, 3, 9, 27} = {1, 3, 9, 27}
  5. \(P \cap F\) = {3}

\(\therefore\) \(U\) = {1, 3, 5, 7, 9, 11, 13, 27}

\(P'\) = {1, 9, 27}

\(F'\) = {5, 7, 11, 13}

\(P'  \cup F\)  = {1, 3, 9, 27}

\(P  \cap F\) = {3}

Example 4

 

 

Assume the Universal set includes 120 students who took a math test.

Out of which, set \(M\) includes 65 students who scored lesser than or equal to 60%.

Set \(N\) includes 25 students who scored lesser than or equal to 40%.

How many students will be in the complement of set \(M\)?

Solution

The above information could be represented in the Venn diagram as:

Venn diagram

Given \(M\) = 65 students who scored lesser than or equal to 60%

\(N\) will be a subset of \(M\).

\(M\)' = Complement of set \(M\) = Students who scored more than 60%

In this case, \(M\) + \(M\)' = \(U\)

Here, \(U\) = Total students = 120

65 + \(P\)' = 120

\(M\)' = 120 - 65 = 55

\(\therefore\) 55 students will be present in the complement of \(M\).

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of the Venn diagram. The math journey around the Venn diagram starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it problems, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions

1. What is a Venn diagram with three circles called?

It is called a 3-circle Venn diagram. It uses three overlapping circles and depicts how the elements of the three sets are related.

2. What does the E or \(\xi\) mean in Venn diagrams?

E or \(\xi\) symbol is generally used to represent a larger set or a universal set.

3. How are Venn diagrams used in our everyday lives?

Venn diagrams are used in many aspects of life.

They are used to categorize or group items, as well as to represent common and uncommon items in various sets.

It is used frequently in statistics, marketing, logic, probability, etc.

  
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