# Set Theory

In this chapter we will learn the basics of set theory with representation, properties and examples.

## What is a set ?

Set is a collection of elements.

The elements can be number, letters, words or symbols.

You can imagine a set as a container which contain different elements.

### Examples of Set

Some common examples of set are;

(i) N = { 1, 2, 3, 4, 5 }

Given above is the set named N which contain numbers 1, 2, 3, 4 & 5.

(ii) X = { a, e, i, o, u }

Given above is the set X which contain vowels of English alphabet a, e, i, o & u.

(iii) Z = { Football, Cricket, Basketball, Tennis }

Given above is the set named Z which contain name of 4 sport.

### Important Points about set representation

Given below are some basic points related to set representation;

(a) Set name is always denoted by capital letters.

In the above examples, the set name is denoted by capital letter alphabets N, X and Z.

(b) The elements of the set are also called member or object in set theory.

(c) In the set, the elements are always enclosed inside the curved bracket { }

(d) If an entity “a” is an element of set A, we can write ” a ϵ A “

a ϵ A tells that entity a belongs to set A, hence ” a ” is an element of set A.

On the other hand, a \cancel{\upepsilon } signifies that the entity ” a ” is not part of set A.

For example, consider the below set X.

X = { 2, 4, 6, 8, 10 }

You can see that number 8 is an element of set X.

So we can write 8 ϵ X, which says ” entity 8 belongs to X “

## Introduction to set theory – Solved Problems

(01) Identify which of the following are examples of set.

(i) Collection of all prime numbers below 20

(ii) Collection of all dangerous animals

(iii) List of all novels written by Dan Brown

(iv) Collection of best novels of Dan Brown

(v) Name of all boys in your class.

Solution
Remember that set is a collection of well defined objects. If the elements are not well defined, the set cannot be formed.

(i) Collection of all prime numbers below 20

It’s an example of set as all the elements inside can be easily defined.

The above set can be represented as;
P = { 2, 3, 5, 7, 11, 13, 17, 19 }

(ii) Collection of all dangerous animals

This is not a set as we cannot define the dangerous elements.

Dangerous animal is a subjective term as some people can find an animal dangerous while other may not.

As the list of dangerous animal will be different for different person, the above collection is not a set.

(iii) List of all novels written by Dan Brown

It’s an example of set as we can easily list the novel name written by Dan Brown.

(iv) Collection of best novels of Dan Brown

Listing “best novels” is again a subjective matter. Here the list change as per the man’s preference.

Hence, the above collection is not a set.

(v) Name of all boys in your class

The collection is a set as the name of boys in particular class is well defined.

(02) Given below is the set A.
A = {1, 3, 5, 7, 9, 11, 13 , 15 }

Insert the symbol 𝜖 or \mathtt{\upepsilon } in the below blank spaces.

(i) 2 . . . A
(ii) 11 . . . A
(iii) 15 . . . A
(iv) 8 . . . A
(iv) 7 . . . A

Solution

(i) 2 \mathtt{\upepsilon } A

Number 2 is not part of set A.

(ii) 11 𝜖 A

Number 11 is an element of set A.

(iii) 15 𝜖 A

Number 15 is an element of set A.

(iv) 8 \mathtt{\upepsilon } A

Number 8 does not belongs to set A

(v) 7 𝜖 A

Number 7 is an element of set A.