Set Theory and Venn Diagram Maths: Formulas, Two and Three Set Problems with Solved Examples for SSC

Set theory is the language of mathematics, and Venn-diagram problems on two or three sets are a regular feature of SSC maths. This guide on set theory and Venn diagrams for SSC explains the types of sets, the operations of union, intersection and difference, the two-set and three-set formulas, subsets and the power set, with solved examples, tricks and practice questions.

What Is a Set?

A set is a well-defined collection of objects, called elements. It is written in curly brackets, for example A = {1, 2, 3}. The number of elements in a set is written n(A).

Types of Sets

Set Operations

Important Formulas

Solved Examples

Example 1. In a class of 60 students, 35 like tea, 30 like coffee and 10 like both. How many like neither?
n(T ∪ C) = 35 + 30 − 10 = 55. Neither = 60 − 55 = 5.

Example 2. Of 100 students, 50 study Maths (M), 40 study Physics (P), 30 study Chemistry (C), 15 study M and P, 10 study P and C, 12 study M and C, and 5 study all three. Find how many study at least one subject, none, and only Maths.
Union = 50 + 40 + 30 − 15 − 10 − 12 + 5 = 88. None = 100 − 88 = 12. Only Maths = 50 − 15 − 12 + 5 = 28.

Example 3. A set has 4 elements. Find the number of subsets, proper subsets and non-empty proper subsets.
Subsets = 2^4 = 16. Proper subsets = 15. Non-empty proper subsets = 14.

Example 4. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find A ∪ B, A ∩ B and A − B.
A ∪ B = {1, 2, 3, 4, 5, 6}; A ∩ B = {3, 4}; A − B = {1, 2}.

Example 5. If n(A) = 20, n(B) = 25 and n(A ∪ B) = 38, find n(A ∩ B).
20 + 25 − 38 = 7.

Example 6. In a group, 70% read newspaper X and 60% read Y, and every person reads at least one. What percentage read both?
70 + 60 − 100 = 30%.

How to Solve Venn Diagram Questions

1. Draw circles for each set inside a rectangle for the universal set.
2. Fill the centre (all three) first, then the two-set regions, then the single-set regions.
3. Subtract overlaps carefully to avoid double counting.
4. Use the total to find the number outside all circles.

Tricks

• Start filling the Venn diagram from the innermost region.
• Use the total percentage as 100 if the data is given in percentages.
• If everyone belongs to at least one set, the union equals the total.
• Number of subsets doubles with each extra element.

Quick Tips

• Fill the centre of the Venn diagram first.
• Do not count the common part twice.
• Use the 2^n rule for subsets.
• Use 100 as the total in percentage problems.

Quick Revision Points

• n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
• Three-set formula has plus triple intersection at the end.
• Number of subsets = 2^n.
• Proper subsets = 2^n − 1.
• De Morgan: (A ∪ B)′ = A′ ∩ B′.

Practice Questions With Answers

1. If n(A) = 15, n(B) = 12 and n(A ∩ B) = 5, find n(A ∪ B).
   Answer: 22
2. How many subsets does a set with 5 elements have?
   Answer: 32
3. How many proper subsets does a set with 3 elements have?
   Answer: 7
4. In a class of 50, 30 play cricket and 25 play football, and all play at least one. How many play both?
   Answer: 5
5. If A = {2, 4, 6, 8} and B = {4, 8, 12}, find A ∩ B.
   Answer: {4, 8}
6. If n(U) = 80, n(A) = 30, n(B) = 40 and n(A ∩ B) = 10, find the number in neither.
   Answer: 20
7. What is the value of n(A − B) if n(A) = 18 and n(A ∩ B) = 6?
   Answer: 12
8. What is the number of non-empty proper subsets of a set with 4 elements?
   Answer: 14
9. If A ∩ B is empty, what are A and B called?
   Answer: Disjoint sets
10. In a survey, 60% like tea and 50% like coffee, and 20% like neither. What percentage like both?
    Answer: 30%

How to Practise This Chapter

Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Quantitative Aptitude notes page, read the previous topic Logarithms and Exponents and the next topic Races and Circular Tracks.

Frequently Asked Questions

What is a set?

A set is a well-defined collection of distinct objects called elements.

What is the formula for two sets?

n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

What is the formula for three sets?

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

What is the universal set?

It is the set that contains all the elements under discussion.

What is the union of sets?

The set of all elements that are in at least one of the sets.

What is the intersection of sets?

The set of elements that are common to the sets.

How many subsets does a set with n elements have?

2^n subsets.

What is a power set?

The set of all subsets of a set. It has 2^n elements.

What are disjoint sets?

Sets that have no element in common.

What is the complement of a set?

The set of elements of the universal set that are not in the given set.

Are Venn diagram questions asked in SSC?

Yes. They are asked in both maths and reasoning sections.

What are common mistakes in set problems?

Counting the common region twice and forgetting those outside all sets.