Set Theory and Venn Diagram – General Aptitude for GATE, PSU, ESE, AE/JE | Priyanka Ma’am

Set Theory and Venn Diagram is an important topic in General Aptitude and is frequently asked in GATE, PSU, ESE, AE/JE exams. It helps in solving problems related to grouping, classification, and logical relationships.

Basic Concepts

A Set is a collection of well-defined objects.

• Finite Set: Limited number of elements
• Infinite Set: Unlimited elements
• Empty Set (Φ): No elements

Important Terminology

• Union (A ∪ B): Elements in A or B or both
• Intersection (A ∩ B): Common elements in A and B
• Complement (A'): Elements not in A
• Universal Set (U): All possible elements

Venn Diagram Concept

Venn diagrams are used to visually represent sets and their relationships using circles.

• Overlapping area → Common elements
• Non-overlapping → Unique elements

Important Formulas

• n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
• n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C)

Key Concepts

1. Two Set Problem

Use:

Total = Only A + Only B + Both

2. Three Set Problem

Divide diagram into 7 regions and fill step by step.

3. Complement Rule

n(A') = Total − n(A)

Solved Examples

Example 1

In a class, 30 students like Maths, 20 like English, and 10 like both. Find students who like at least one.

n(A ∪ B) = 30 + 20 − 10 = 40

Example 2

Out of 100 students, 60 like tea, 50 like coffee, and 20 like both. Find students who like neither.

n(A ∪ B) = 60 + 50 − 20 = 90
Neither = 100 − 90 = 10

Example 3

If total = 80 and 50 are in set A, find complement.

A' = 80 − 50 = 30

Example 4 (GATE Level)

In a survey of 200 people: 120 like cricket, 100 like football, 60 like both. Find people who like only cricket.

Only cricket = 120 − 60 = 60

Shortcut Tricks

• Always draw Venn diagram before solving
• Fill intersection first
• Use formulas instead of guesswork
• Check total at the end

Questions

Q1. n(A)=40, n(B)=30, n(A∩B)=10. Find n(A∪B).

= 40 + 30 − 10 = 60

Q2. Total = 100, n(A)=70. Find n(A').

= 100 − 70 = 30

Exam Tips

• Never skip diagram
• Be careful with “at least” and “only” terms
• Practice 3-set problems regularly
• Verify totals to avoid mistakes

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