Venn Diagram

Venn diagrams are visual tools used to illustrate the logical relationships between different sets of items. They use overlapping circles to show how sets intersect, share elements, or remain distinct. By categorizing items in this way, Venn diagrams make it easier to understand similarities and differences.

Venn diagrams are also known as set diagrams or logical diagrams.

Terms Related to Venn Diagram

The concept of the Venn diagram is very useful for solving a variety of problems in Mathematics and other areas. To understand more about it, let's learn some important terms related to it.

1) Set

A set is a collection of well-defined, distinct objects, considered as a single entity.

Example: A set of natural numbers: {1, 2, 3, 4, 5, 6, ...}

2) Universal Set

Universal Set is a large set that contains all the sets that we are considering in a particular situation.

For example, suppose we are considering the set of Honda cars in a society say set A, and let set B is the group of red car in the same society then the set of all the cars in that society is the universal set as it contains the values of both the sets, set A and set B in consideration.

3) Subset

Subset is actually a set of values that is contained inside another set i.e. we can say that set B is the subset of set A if all the values of set B are contained in set A.

For example, if we take N as the set of all the natural numbers and W as the set of all whole numbers then,

• N = Set of all Natural Numbers
• W = Set of all Whole Numbers

We can say that N is a subset of W all the values of set N are contained in set W i.e. N ⊆ W

4) Disjoint Sets

Disjoint sets are sets that do not have any elements in common. If none of the elements of set A are present in set B, then sets A and B are disjoint. In other words, the intersection of two disjoint sets is an empty set.

For example, let A be the set of even numbers less than 10 and B be the set of odd numbers less than 10. Then,
A = {2, 4, 6, 8}
B = {1, 3, 5, 7, 9}
We can say that A and B are disjoint sets because they do not share any elements, i.e., A ∩ B = ∅.

Symbols

Some of the most important types of symbols used in drawing Venn diagrams are,

Steps To Draw Venn Diagram

Step 1: Identify the sets

• Decide how many sets you are representing (1, 2, 3, or more).
• Example: Set A = even numbers, Set B = prime numbers.

Step 2: Draw the Universal Set

• Draw a rectangle to represent the universal set (U) containing all possible elements in the problem.

Step 3: Draw Circles for Each Set

• Draw one circle for each set inside the rectangle.
• Label each circle clearly (A, B, C…).
• Make the circles overlap if the sets have common elements.

Step 4: Fill in the Elements

• Intersection areas: Place elements that belong to more than one set in the overlapping regions.
• Exclusive areas: Place elements that belong to only one set in the non-overlapping parts of the circle.
• Outside all circles: Place elements that do not belong to any set in the rectangle outside the circles.

Example: Take a set A representing even numbers up to 10 and another set B representing natural numbers less than 5 then their interaction is represented using the Venn diagram.

Venn Diagram for Sets Operations

There are different operations that can be done on sets in order to find the possible unknown parameter, for example, if two sets have something in common, their intersection is possible. The basic operations performed on the set are,

1. Venn Diagram of Union of Sets

The Union of two or more two sets represents the data of the sets without repeating the same data more than once, it is shown with the symbol ⇢∪.

n(A∪ B) = {a: a∈ A OR a∈ B}

2. Venn Diagram of Intersection of Sets

The intersection of two or more two sets means extracting only the amount of data that is common between/among the sets. The symbol used for the intersection⇢ ∩.

n(A∩ B) = {a: a∈ A and a∈ B}

3. Venn Diagram of Complement of a Set

Complementing a set means finding the value of the data present in the Universal set other than the data of the set.

n(A') = U- n(A)

4. Venn Diagram of Difference of Set

Suppose we take two sets, Set A and Set B then their difference is given as A - B. This difference represents all the values of set A which are not present in set B.

For example, if we take Set A = {1, 2, 3, 4, 5, 6} and set B = {2, 4, 6, 8} then A- B = {1, 3, 5}. 

In the Venn diagram, we represent the A - B as the area of set A which is not intersecting with set B.

Three-Set Venn Diagram

We can represent three sets easily using the Venn Diagram. Their representation is done by three overlapping circles. Suppose we take three sets of Set A of the people who play cricket. Set B of the people who are graduates and Set C of the people who are 18 years and above of the age. 

Then the Venn diagram representing the above three sets is drawn using three circles and taking their intersection wherever required.

We can represent the intersection of three sets using the Venn diagram. The below image represents the intersection of three sets.

We can find the various parameters using the above Venn diagram.

Suppose we have to find,

• No of graduates who play cricket it is given by B⋂C
• No of graduates who play cricket and are at least 18 years old is given by A⋂B⋂C, etc.

Example: In a class of 40 students, 18 like Mathematics, 16 like Science, and 10 like both Mathematics and Science. Then find the students who like either Mathematics or Science.

Solution:

Let A be the set of students who like Mathematics and B be the set of students who like Science, then
n(A) = 18,
n(B) = 16, and

n(A ⋂ B) = 10

Now to find the number of students who like either Mathematics or Science i.e. n(A U B) we use the above formula.

n(A U B) = n(A) + n(B) – n (A ⋂ B)
⇒ n(A U B) = 18 + 16 - 10
⇒ n(A U B) = 24

Practice Questions on Venn Diagrams