Venn Diagrams: Definition, Types, Properties & Applications in JEE 

1.0What is a Venn Diagram?

A Venn diagram is a powerful graphical representation used to show the relationships between different sets. Also known as a set diagram or a logic diagram, it was introduced by John Venn as a way to visually describe similarities and differences between collections of items. These diagrams typically use overlapping circles to depict the logical relationships between sets, while a rectangle is used to represent the universal set, which contains all the elements under consideration.

The true power of Venn diagrams lies in their ability to visually represent set operations and their outcomes, which is the key to solving a wide range of problems in mathematics, logic, statistics, and computer science.

2.0Terms Related to Venn Diagram

Universal Set (U)

This is the large rectangle that encompasses all the elements being discussed in a particular context. It represents the "universe" of all possible elements for a given problem. Every set within the diagram is considered a subset of the universal set.

Subset

A subset is a set that is entirely contained within another set. In a Venn diagram, this is represented by a circle placed completely inside another circle, indicating that every element of the smaller set is also an element of the larger set. 

3.0Venn Diagram for Sets Operations

a) Union of Sets (A∪B)

The union of two sets A and B, denoted as A∪B, is the set of all elements that are in set A, or in set B, or in both. Think of it as combining all the elements from both sets. The word "or" is key here.

• Venn Diagram: The union is represented by the total shaded area of both circles.

b) Intersection of Sets (A∩B)

The intersection of two sets A and B, denoted as A∩B, is the set of elements that are common to both A and B. The word "and" is the key to remembering this operation.

• Venn Diagram: The intersection is the overlapping region where the two circles meet. If two sets have no common elements, they are called disjoint sets, and their circles do not overlap.

c) Difference of Sets (A−B)

The difference of set A and B, denoted as A−B or A∖B, is the set of all elements that are in set A but not in set B.

• Venn Diagram: This is the portion of circle A that does not overlap with circle B.
• Note: A−B is not the same as B−A. The latter is the part of circle B that does not overlap with A.

d) Complement of a Set (A′ or Ac)

The complement of a set A, denoted as A′ or Ac, is the set of all elements in the universal set (U) that are not in set A.

• Venn Diagram: The complement is the area of the rectangle outside of circle A.

4.0Types of Venn Diagrams

(i) Two-Set Venn Diagram

This is the most common and simplest type of Venn diagram. It consists of two overlapping circles inside a rectangle. It is used to illustrate relationships and operations between two distinct sets, such as union, intersection, and difference.

(ii) Three-Set Venn Diagram

This type of diagram uses three overlapping circles to represent the relationships between three sets. This is a crucial diagram for JEE as it is often used in word problems involving three different categories, subjects, or groups. The diagram is divided into eight distinct regions, including the central region representing the intersection of all three sets.

(iii) Four-Set and Beyond Venn Diagram

Venn diagrams can be extended to four or more sets, but they become highly complex and difficult to draw in a way that shows all possible intersections. A four-set diagram can be drawn using four intersecting ellipses. For five or more sets, the diagrams are typically abstract and not practical for visual problem-solving. In such cases, the Inclusion-Exclusion Principle is used directly without a visual aid.

5.0Representation of Sets using Venn Diagrams

Example 1:
Let U = {1, 2, 3, 4, 5, 6}, A = {2, 4, 6}, B = {1, 2, 3}.

• A ∩ B = {2}
• A ∪ B = {1, 2, 3, 4, 6}
• A′ = {1, 3, 5}

These can be represented in a two-circle Venn diagram inside the rectangle U.

Example 2 (Three Sets):
Out of 100 students:

• 40 like Mathematics
• 50 like Physics
• 30 like Chemistry
• 10 like all three subjects
  Using a three-circle Venn diagram, students can calculate union, intersection, and exclusive areas.

6.0Properties of Venn Diagrams

Laws of Set Theory Visualized with Venn Diagrams

1. Commutative Law:
2. • A ∪ B = B ∪ A
   • A ∩ B = B ∩ A
2. Associative Law:
3. • (A ∪ B) ∪ C = A ∪ (B ∪ C)
   • (A ∩ B) ∩ C = A ∩ (B ∩ C)
3. Distributive Law:
4. • A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

De Morgan’s Laws with Venn Diagrams

• (A ∪ B)′ = A′ ∩ B′
• (A ∩ B)′ = A′ ∪ B′

These are best understood visually with shaded Venn diagrams.

7.0Common Applications of Venn Diagrams in JEE

1. Probability:
2. • Example: If P(A) = 0.5, P(B) = 0.3, and P(A ∩ B) = 0.2, then
     P(A ∪ B) = P(A) + P(B) – P(A ∩ B) = 0.6.
2. Set Operations in JEE:
3. • Used for solving questions on relations, functions, and mappings.
3. Logical Reasoning & Puzzles:
4. • Many aptitude-based JEE questions rely on set representation via Venn diagrams.

8.0Step-by-Step Method to Solve Venn Diagram Questions

1. Identify the universal set and total elements.
2. Write down given values (individual, intersections, unions).
3. Draw circles and fill overlapping regions first.
4. Use subtraction to fill non-overlapping regions.
5. Cross-check with total count to ensure correctness.

9.0Solved Examples on Venn Diagram

Q1. In a class of 60 students, 25 play Cricket, 20 play Football, 15 play both. Find how many students play at least one game.

Solution:
n(C ∪ F) = n(C) + n(F) – n(C ∩ F)
= 25 + 20 – 15 = 30

So, 30 students play at least one game.

Q2. Out of 100 students, 60 study Physics, 45 study Chemistry, 25 study both. Find how many study neither subject.

Solution:
n(P ∪ C) = 60 + 45 – 25 = 80
n(neither) = 100 – 80 = 20

Q3. A survey shows that 50 students like Mathematics, 30 like Physics, 20 like Chemistry, and 10 like all three. Find the number of students who like exactly two subjects.

Solution (Three-set Venn):
Use formula:
n(exactly two) = [n(M ∩ P) + n(P ∩ C) + n(C ∩ M)] – 3n(M ∩ P ∩ C)

10.0Practice Questions on Venn Diagram

1. Draw a Venn diagram to represent the following:
2. • (i) ( A = {1, 2, 3} ), ( B = {3, 4, 5} )
   • (ii) ( P = {a, e, i, o, u} ), ( Q = {a, b, c, d, e} )
2. In a group of 50 students, 30 study Mathematics, 25 study Physics, and 10 study both. Represent this information using a Venn diagram and find the number of students who study:
3. • (i) only Mathematics
   • (ii) only Physics
   • (iii) neither Mathematics nor Physics
3. Out of 100 students, 60 play cricket, 45 play football, and 25 play both cricket and football. Represent this data in a Venn diagram and find:
4. • (i) Number of students who play only cricket
   • (ii) Number of students who play only football
   • (iii) Number of students who play at least one game
4. Let ( U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} ), ( A = {2, 4, 6, 8, 10} ), and ( B = {3, 6, 9} ).
5. • Draw a Venn diagram for ( A ) and ( B ).
   • Find $(A\cup B),(A\cap B)$, and ( A' ) (complement of ( A )).
5. A survey shows that out of 120 people, 80 like tea, 50 like coffee, and 30 like both. Represent this using a Venn diagram and find how many people like:
6. • (i) only tea
   • (ii) only coffee
   • (iii) neither tea nor coffee
6. Prove De Morgan’s law $((A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime })$ using a Venn diagram.
7. In a class of 40 students, 18 take Mathematics, 20 take Biology, and 7 take both. Represent the data with a Venn diagram and find how many students take neither subject.
8. A group of 200 students were asked about their preference for three activities: music, dance, and painting. If 80 like music, 100 like dance, 70 like painting, 30 like both music and dance, 20 like both dance and painting, 25 like both music and painting, and 10 like all three, represent the data using a three-set Venn diagram and answer related questions.
9. Verify the law $(A\cap (B\cup C)=(A\cap B)\cup (A\cap C))$ with the help of a Venn diagram.
10. Solve a logical puzzle using a Venn diagram: In a class, 15 students speak Hindi, 12 speak English, and 8 speak both. Draw a Venn diagram and find how many students speak at least one language.