Tricks and Problems on SET Theory


Set theory has its own notations and symbols that can seem unusual for many. In this tutorial, we look at some solved examples to understand how set theory works and the kind of problems it can be used to solve.


A set is a collection of objects.

It is usually represented in flower braces.

For example:
Set of natural numbers           = {1,2,3,…..}
Set of whole numbers             = {0,1,2,3,…..}

Each object is called an element of the set.

The set that contains all the elements of a given collection is called the universal set and is represented by the symbol ‘µ’, pronounced as ‘mu’.

For two sets A and B,
n(A ᴜ B) is the number of elements present in either of the sets A or B.
n(A ∩ B) is the number of elements present in both the sets A and B.
n(AᴜB) = n(A) + (n(B) – n(A∩B)



For three sets A, B and C,

n(AᴜBᴜC) = n(A) + n(B) + n(C) – n(A∩B) – n(B∩C) – n(C∩A) + n(A∩B∩C)

The intersection of sets is only those elements common to all sets. Let’s call our sets A, B, and C. If ‘∩ = intersection‘ and ‘ᴜ = union‘, the need-to-know formulas are:

P(A U B U C) = P(A) + P(B) + P(C) – P(A B) – P(A C) – P(B C) + P(A B C)

To find the number of people in exactly one set:

P(A) + P(B) + P(C) – 2P(A B) – 2P(A C) – 2P(B C) + 3P(A B C)

To find the number of people in exactly two sets:

P(A  B) + P(A  C) + P(B  C) – 3P(A  B  C)

To find the number of people in exactly three sets:

P(A  B  C)

To find the number of people in two or more sets:

P(A  B) + P(A  C) + P(B  C) – 2P(A  B  C)

To find the number of people in at least one set:

P(A) + P(B) + P(C) – P(A  B) – P(A  C) – P(B  C) + 2 P(A  B ∩ C)


In a class of 100 students, 35 like science and 45 like math. 10 like both. How many like neither?

Correct! Wrong!

There are 30 students in a class. Among them, 8 students are learning both English and French. A total of 18 students are learning English. If every student is learning at least one language, how many students are learning French in total?

Correct! Wrong!

In a group, there were 115 people whose proofs of identity were being verified. Some had passport, some had voter id and some had both. If 65 had passport and 30 had both, how many had voter id only and not passport?

Correct! Wrong!

Among a group of people, 40% liked red, 30% liked blue and 30% liked green. 7% liked both red and green, 5% liked both red and blue, 10% liked both green and blue. If 86% of them liked at least one colour, what percentage of people liked all three?

Correct! Wrong!

Let A = {0, 1, 3, 5}, B = {5, 6, 1, 3, 9} and C = {0, 1, 2, 3, 9, 13}. Then, (A&B) OR C :

Correct! Wrong!

set theory
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