Venn Diagrams

1. What are Venn Diagrams?

Venn Diagrams are visual representations that show logical relationships between different sets using overlapping circles. They help in understanding set theory and logical connections.

Visual Representation

Shows Set Relationships

Uses Overlapping Circles

Logical Problem Solving

Simple Definition: Venn Diagrams are graphical representations that use circles to show all possible logical relations between a finite collection of different sets.

Basic Venn Diagram Types

Common Venn Diagram Structures

Type	Description	Diagram	Example
Two Set Diagram	Two overlapping circles	○ ○ (overlapping)	Doctors & Engineers
Three Set Diagram	Three overlapping circles	Three ○ overlapping	Doctors, Engineers, Teachers
Disjoint Sets	No overlapping circles	○ ○ ○	Different categories
Subset	One circle inside another	○ inside ◯	Doctors ⊂ Medical Professionals
Universal Set	Rectangle enclosing circles	□ with circles inside	All people in a company

2. Set Operations in Venn Diagrams

Key Set Operations

Basic Operations

Union (A ∪ B)

All elements in A or B or both

Intersection (A ∩ B)

Elements common to both A and B

Complement (A')

Elements not in A

Advanced Operations

Difference (A - B)

Elements in A but not in B

Symmetric Difference

Elements in A or B but not both

Universal Set (U)

All possible elements

3. How to Solve Venn Diagram Problems?

Follow these step-by-step methods to solve Venn diagram questions:

1

Identify Sets

Determine what each circle represents

2

Draw Diagram

Create circles with overlapping regions

3

Fill Information

Start from intersection areas

4

Calculate Missing Values

Use given totals to find unknown values

4. Three Set Venn Diagram Regions

Regions in Three Overlapping Sets

Region	Description	Notation	Example (A, B, C)
Only A	Elements only in A	A - (B ∪ C)	Only Doctors
A ∩ B only	Only in A and B, not in C	(A ∩ B) - C	Doctors & Engineers but not Teachers
A ∩ B ∩ C	Common to all three	A ∩ B ∩ C	Doctors, Engineers & Teachers
Only B	Elements only in B	B - (A ∪ C)	Only Engineers
B ∩ C only	Only in B and C, not in A	(B ∩ C) - A	Engineers & Teachers but not Doctors
Only C	Elements only in C	C - (A ∪ B)	Only Teachers
A ∩ C only	Only in A and C, not in B	(A ∩ C) - B	Doctors & Teachers but not Engineers
None	Not in any set	U - (A ∪ B ∪ C)	Not Doctor, Engineer or Teacher

5. Solved Examples

Example 1: Two Set Problem

In a class of 50 students, 30 play cricket, 25 play football, and 10 play both. How many play neither game?

Step 1: Let C = Cricket, F = Football

Step 2: Only Cricket = 30 - 10 = 20

Step 3: Only Football = 25 - 10 = 15

Step 4: Total playing games = 20 + 15 + 10 = 45

Step 5: Neither = Total - Playing = 50 - 45 = 5

Answer: 5 students play neither game

Example 2: Three Set Problem

In a survey of 100 people: 60 read Times, 50 read Hindu, 40 read Express, 30 read Times & Hindu, 20 read Hindu & Express, 15 read Times & Express, and 10 read all three. How many read exactly one newspaper?

Step 1: Only Times & Hindu = 30 - 10 = 20

Step 2: Only Hindu & Express = 20 - 10 = 10

Step 3: Only Times & Express = 15 - 10 = 5

Step 4: Only Times = 60 - (20+10+5) = 25

Step 5: Only Hindu = 50 - (20+10+10) = 10

Step 6: Only Express = 40 - (5+10+10) = 15

Step 7: Exactly one = 25 + 10 + 15 = 50

Answer: 50 people read exactly one newspaper

Example 3: Logical Venn Diagram

Which diagram represents the relationship: Doctors, Surgeons, Professionals?

Step 1: All Surgeons are Doctors

Step 2: All Doctors are Professionals

Step 3: Therefore, Surgeons ⊂ Doctors ⊂ Professionals

Answer: Three concentric circles with Surgeons inside Doctors inside Professionals

6. Important Formulas

Venn Diagram Formulas

Two Set Formulas

n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

n(Only A) = n(A) - n(A ∩ B)

n(Neither) = Total - n(A ∪ B)

Three Set Formulas

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A∩B) - n(B∩C) - n(A∩C) + n(A∩B∩C)

n(Exactly one) = n(only A) + n(only B) + n(only C)

n(Exactly two) = n(A∩B only) + n(B∩C only) + n(A∩C only)

7. Quick Recap

Concept	Symbol	Meaning	Example
Union	A ∪ B	All elements in A or B	Doctors or Engineers
Intersection	A ∩ B	Common elements	Doctor Engineers
Complement	A'	Not in A	Not Doctors
Difference	A - B	In A but not in B	Doctors but not Engineers
Universal Set	U	All elements	All people surveyed

8. Delhi Police Exam Tips

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Always draw diagrams - visual representation helps avoid mistakes

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Start from intersection - fill common areas first

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Memorize key formulas - especially for three sets

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Check your work - ensure all regions add up to total

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Practice logical relationships - subset, disjoint sets