Venn Diagram

A Venn diagram is used to visually represent the differences and the similarities between two concepts. Venn diagrams are also called logic or set diagrams and are widely used in set theory, logic, mathematics, businesses, teaching, computer science, and statistics.

Let's learn about Venn diagrams, their definition, symbols, and types with solved examples.

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.

Venn Diagram Example

Let us observe a Venn diagram example. Here is the Venn diagram that shows the correlation between the following set of numbers.

• One set contains even numbers from 1 to 25 and the other set contains the numbers in the 5x table from 1 to 25.
• The intersecting part shows that 10 and 20 are both even numbers and also multiples of 5 between 1 to 25.

Terms Related to Venn Diagram

Let us understand the following terms and concepts related to Venn Diagram, to understand it better.

Universal Set

Whenever we use a set, it is easier to first consider a larger set called a universal set that contains all of the elements in all of the sets that are being considered. Whenever we draw a Venn diagram:

• A large rectangle is used to represent the universal set and it is usually denoted by the symbol E or sometimes U.
• All the other sets are represented by circles or closed figures within this larger rectangle.
• Every set is the subset of the universal set U.

Consider the above-given image:

• U is the universal set with all the numbers 1-10, enclosed within the rectangle.
• A is the set of even numbers 1-10, which is the subset of the universal set U and it is placed inside the rectangle.
• All the numbers between 1-10, that are not even, will be placed outside the circle and within the rectangle as shown above.

Subset

Venn diagrams are used to show subsets. A subset is actually a set that is contained within another set. Let us consider the examples of two sets A and B in the below-given figure. Here, A is a subset of B. Circle A is contained entirely within circle B. Also, all the elements of A are elements of set B.

This relationship is symbolically represented as A ⊆ B. It is read as A is a subset of B or A subset B. Every set is a subset of itself. i.e. A ⊆ A. Here is another example of subsets :

• N = set of natural numbers
• I = set of integers
• Here N ⊂ I, because all-natural numbers are integers.

There are more than 30 Venn diagram symbols. We will learn about the three most commonly used symbols in this section. They are listed below as:

Let us understand the concept and the usage of the three basic Venn diagram symbols using the image given below.

In set theory, we can perform certain operations on given sets. These operations are as follows,

• Union of Set
• Intersection of set
• Complement of set
• Difference of set

Union of Sets Venn Diagram

The union of two sets A and B can be given by: A ∪ B = {x | x ∈ A or x ∈ B}. This operation on the elements of set A and B can be represented using a Venn diagram with two circles. The total region of both the circles combined denotes the union of sets A and B.

Intersection of Set Venn Diagram

The intersection of sets, A and B is given by: A ∩ B = {x : x ∈ A and x ∈ B}. This operation on set A and B can be represented using a Venn diagram with two intersecting circles. The region common to both the circles denotes the intersection of set A and Set B.

Complement of Set Venn Diagram

The complement of any set A can be given as A'. This represents elements that are not present in set A and can be represented using a Venn diagram with a circle. The region covered in the universal set, excluding the region covered by set A, gives the complement of A.

Difference of Set Venn Diagram

The difference of sets can be given as, A - B. It is also referred to as a ‘relative complement’. This operation on sets can be represented using a Venn diagram with two circles. The region covered by set A, excluding the region that is common to set B, gives the difference of sets A and B.

We can observe the above-explained operations on sets using the figures given below,

Three sets Venn diagram is made up of three overlapping circles and these three circles show how the elements of the three sets are related. When a Venn diagram is made of three sets, it is also called a 3-circle Venn diagram. In a Venn diagram, when all these three circles overlap, the overlapping parts contain elements that are either common to any two circles or they are common to all the three circles. Let us consider the below given example:

Here are some important observations from the above image:

• Elements in P and Q = elements in P and Q only plus elements in P, Q, and R.
• Elements in Q and R = elements in Q and R only plus elements in P, Q, and R.
• Elements in P and R = elements in P and R only plus elements in P, Q, and R.

Venn diagrams can be drawn with unlimited circles. Since more than three becomes very complicated, we will usually consider only two or three circles in a Venn diagram. Here are the 4 easy steps to draw a Venn diagram:

• Step 1: Categorize all the items into sets.
• Step 2: Draw a rectangle and label it as per the correlation between the sets.
• Step 3: Draw the circles according to the number of categories you have.
• Step 4: Place all the items in the relevant circles.

Example: Let us draw a Venn diagram to show categories of outdoor and indoor for the following pets: Parrots, Hamsters, Cats, Rabbits, Fish, Goats, Tortoises, Horses.

• Step 1: Categorize all the items into sets (Here, its pets): Indoor pets: Cats, Hamsters, and, Parrots. Outdoor pets: Horses, Tortoises, and Goats. Both categories (outdoor and indoor): Rabbits and Fish.
• Step 2: Draw a rectangle and label it as per the correlation between the two sets. Here, let's label the rectangle as Pets.
• Step 3: Draw the circles according to the number of categories you have. There are two categories in the sample question: outdoor pets and indoor pets. So, let us draw two circles and make sure the circles overlap.

• Step 4: Place all the pets in the relevant circles. If there are certain pets that fit both the categories, then place them at the intersection of sets, where the circles overlap. Rabbits and fish can be kept as indoor and outdoor pets, and hence they are placed at the intersection of both circles.

• Step 5: If there is a pet that doesn't fit either the indoor or outdoor sets, then place it within the rectangle but outside the circles.

For any two given sets A and B, the Venn diagram formula is used to find one of the following: the number of elements of A, B, A U B, or A ⋂ B when the other 3 are given. The formula says:

• n(A U B) = n(A) + n(B) – n (A ⋂ B)

Here, n(A) and n(B) represent the number of elements in A and B respectively. n(A U B) and n(A ⋂ B) represent the number of elements in A U B and A ⋂ B respectively. This formula is further extended to 3 sets as well and it says:

• n (A U B U C) = n(A) + n(B) + n(C) - n(A ⋂ B) - n(B ⋂ C) - n(C ⋂ A) + n(A ⋂ B ⋂ C)

Here is an example of Venn diagram formula.

Example: In a cricket school, 12 players like bowling, 15 like batting, and 5 like both. Then how many players like either bowling or batting.

Solution:

Let A and B be the sets of players who like bowling and batting respectively. Then

n(A) = 12

n(B) = 15

n(A ⋂ B) = 5

We have to find n(A U B). Using the Venn diagram formula,

n(A U B) = n(A) + n(B) – n (A ⋂ B)

n(A U B) = 12 + 15 - 5 = 22.

There are several advantages to using Venn diagrams. Venn diagram is used to illustrate concepts and groups in many fields, including statistics, linguistics, logic, education, computer science, and business.

• 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.
• 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.

☛Related Articles:

Check out the following pages related to Venn diagrams:

• Operations on Sets
• Roster Notation
• Set Builder Notation
• Statistics
• Probability

Important Notes on Venn Diagrams:

Here is a list of a few points that should be remembered while studying Venn diagrams:

• Every set is a subset of itself i.e., A ⊆ A.
• A universal set accommodates all the sets under consideration.
• If A ⊆ B and B ⊆ A, then A = B
• The complement of a complement is the given set itself.

What is a Venn Diagram in Math?

In math, a Venn diagram is used to visualize the logical relationship between sets and their elements and helps us solve examples based on these sets.

How do You Read a Venn Diagram?

These are steps to be followed while reading a Venn diagram:

• First, observe all the circles that are present in the entire diagram.
• Every element present in a circle is its own item or data set.
• The intersecting or the overlapping portions of the circles contain the items that are common to the different circles.
• The parts that do not overlap or intersect show the elements that are unique to the different circle.

What is the Importance of Venn Diagram?

Venn diagrams are used in different fields including business, statistics, linguistics, etc. Venn diagrams can be used to visually organize information to see the relationship between sets of items, such as commonalities and differences, and to depict the relations for visual communication.

What is the Middle of a Venn Diagram Called?

When two or more sets intersect, overlap in the middle of a Venn diagram, it is called the intersection of a Venn diagram. This intersection contains all the elements that are common to all the different sets that overlap.

How to Represent a Universal Set Using Venn Diagram?

A large rectangle is used to represent the universal set and it is usually denoted by the symbol E or sometimes U. All the other sets are represented by circles or closed figures within this larger rectangle that represents the universal set.

What are the Different Types of Venn Diagrams?

The different types of Venn diagrams are:

• Two-set Venn diagram: The simplest of the Venn diagrams, that is made up of two circles or ovals of different sets to show their overlapping properties.
• Three-set Venn diagram: These are also called the three-circle Venn diagram, as they are made using three circles.
• Four-set Venn diagram: These are made out of four overlapping circles or ovals.
• Five-set Venn diagram: These comprise of five circles, ovals, or curves. In order to make a five-set Venn diagram, you can also pair a three-set diagram with repeating curves or circles.

What are the Different Fields of Applications of Venn Diagrams?

There are different cases of applications of Venn diagrams: Set theory, logic, mathematics, businesses, teaching, computer science, and statistics.

Can a Venn Diagram Have 2 Non Intersecting Circles?

Yes, a Venn digram can have two non intersecting circles where there is no data that is common to the categories belonging to both circles.

What is the Formula of Venn Diagram?

The formula that is very helpful to find the unknown information about a Venn diagram is n(A U B) = n(A) + n(B) – n (A ⋂ B), where

• A and B are two sets.
• n(A U B) is the number of elements in A U B.
• n (A ⋂ B) is the number of elements in A ⋂ B.

Can a Venn Diagram Have 3 Circles?

Yes, a Venn diagram can have 3 circles, and it's called a three-set Venn diagram to show the overlapping properties of the three circles.

What is Union in the Venn Diagram?

A union is one of the basic symbols used in the Venn diagram to show the relationship between the sets. A union of two sets C and D can be shown as C ∪ D, and read as C union D. It means, the elements belong to either set C or set D or both the sets.

What is A ∩ B Venn Diagram?

A ∩ B (which means A intersection B) in the Venn diagram represents the portion that is common to both the circles related to A and B.  A ∩ B can be a null set as well and in this case, the two circles will either be non-intersecting or can be represented with intersecting circles having no data in the intersection portion.