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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.
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 nonintersecting 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 abovegiven image:
 U is the universal set with all the numbers 110, enclosed within the rectangle.
 A is the set of even numbers 110, which is the subset of the universal set U and it is placed inside the rectangle.
 All the numbers between 110, 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 belowgiven 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 allnatural numbers are integers.
Venn Diagram Symbols
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:
Venn Diagram Symbols  Explanation 

The union symbol  ∪ 
A ∪ B is read as A union B. Elements that belong to either set A or set B or both the sets. U is the universal set. 
The intersection symbol  ∩ 
A ∩ B is read as A intersection B. Elements that belong to both sets A and B. U is the universal set. 
The complement symbol  A^{c} or A' 
A' is read as A complement. Elements that don't belong to set A. U is the universal set. 
Let us understand the concept and the usage of the three basic Venn diagram symbols using the image given below.
Symbol  It refers to  Total Elements (No. of students) 

A ∪ C  The number of students that prefer either burger or pizza or both.  1 + 10 + 2 + 2 + 6 + 9 = 30 
A ∩ C  The number of students that prefer both burger and pizza.  2 + 2 = 4 
A ∩ B ∩ 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 
Venn Diagram for Sets Operations
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 aboveexplained operations on sets using the figures given below,
Venn Diagram for Three Sets
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 3circle 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.
How to Draw a Venn Diagram?
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.
Venn Diagram Formula
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.
Applications of Venn Diagram
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.
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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.
Examples of Venn Diagram

Example 1: Let us take an example of a set with various types of fruits, A = {guava, orange, mango, custard apple, papaya, watermelon, cherry}. Represent these subsets using sets notation: a) Fruit with one seed b) Fruit with more than one seed
Solution: Among the various types of fruit, only mango and cherry have one seed.
Thus,
Answer: a) Fruit with one seed = {mango, cherry} b) Fruit with more than one seed = {guava, orange, custard apple, papaya, watermelon}
Note: If we represent these two sets on a Venn diagram, the intersection portion is empty.

Example 2: Let us take an example of two sets A and B, where A = {3, 7, 9} and B = {4, 8}. These two sets are subsets of the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Find A ∪ B.
Solution: The Venn diagram for the above relations can be drawn as:
Answer: A ∪ B means, all the elements that belong to either set A or set B or both the sets = {3, 4, 7, 8, 9}

Example 3: Using Venn diagram, find X ∩ Y, given that X = {1, 3, 5}, Y = {2, 4, 6}.
Solution:
Given: X = {1, 3, 5}, Y = {2, 4, 6}
The Venn diagram for the above example can be given as,
Answer: From the blue shaded portion of Venn diagram, we observe that, X ∩ Y = ∅ (null set).
FAQs on Venn Diagrams
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:
 Twoset 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.
 Threeset Venn diagram: These are also called the threecircle Venn diagram, as they are made using three circles.
 Fourset Venn diagram: These are made out of four overlapping circles or ovals.
 Fiveset Venn diagram: These comprise of five circles, ovals, or curves. In order to make a fiveset Venn diagram, you can also pair a threeset 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 threeset 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 nonintersecting or can be represented with intersecting circles having no data in the intersection portion.
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