Sets is a well-defined collection of distinct objects. It should be readable and easily understandable. In this chapter, we will discuss the definition of sets, elements of sets, and ways to describe a set.

## Summary

A set is always enclosed within the opening and closing of the curly brackets i.e. '{' and '}'. For example: A = {a}.

Elements of sets should always be written in small letters.

No elements in a set can be repeated.

The letter representing a set should always be written in capital form. For example: A = {a} is correct but a = {a} is incorrect way of writing a set.

## Introduction to Sets

Set is a well-defined collection of distinct objects.

Well-defined means a set should be understandable. A set should be easy to describe.

Distinct objects means the members in a set should be unique and should not be repeated.

Example of Set:

A = {a,e,i,o,u}

A = {x: x is a set of vowel letters in English alphabet}

Above set is well-defined because we can easily describe the set and is understandable. Anyone can say that it is a collection of vowels in English alphabets. Similarly, it has distinct and unique elements. None of the elements are identical to each other.

## Element of Sets

Members of sets are known as elements of that sets. Instead of saying member of sets, we prefer saying 'elements of sets'. This is a term to remember for further study of sets.

In a set A = {a,e,i,o,u}, the members of Set A i.e. 'a', 'e', 'i', 'o', and 'u' are said to be its elements.

**Notes**

If the elements of a set are alphabets, they should always be written in small letters.

No elements can be repeated within a single set.

## Ways of Writing a Set

There are many ways to represent a set. Out of all, only two of the most popular ways of writing a set are explained below:

Listing or Roster or Tabular Method

Set Builder's Method

### Listing Method

The listing method is also known as the roster or tabular method. In this method, the elements of a set are listed within two curly brackets (or braces) '{' and '}'.

Example: A = {a,e,i,o,u}

**When is Listing method useful?**

It can be used to represent sets with less number of elements or a set that has a sequence of elements.

When we have to represent a set with only one element, listing method becomes very much useful because it saves time and space, as well.

Similarly, we can use it to represent a large number of elements but they should follow a sequence. For example: A = {1,2,3,.....,97,98,99}. However, this method is not recommended for educational use. Do not follow this method to represent such large number of elements in an examination.

### Set Builder's Method

Set Builder's Method is another way to represent a set in which the elements of a set are represented by variable(s) and value of the variable(s) is/are assigned by using language or mathematical expressions.

Example: A = {x : x is a set of vowel letters of English alphabets}

The set is read as X is such that X is a set of vowel letters of English alphabets. The symbol ':' is read as such that.

How to list elements from a set written using Set Builder's method?

Another example: A = {x: x = 2 + n ; x is natural numbers less than 5}

While writing the elements of this set, we need to loop the given mathematical expression 2 + n for specified number of times to get the distinct elements of the set.

First, we need to know the value of n because that is what value of x depends upon.

Here, n = (1,2,3,4) because it is natural numbers less than 5.

Now,

first element = 2 + n = 2 + 1 = 3

second element = 2 + n = 2 + 2 = 4

third element = 2 + n = 2 + 3 = 5

fourth element = 2 + n = 2 + 4 = 6

So, the elements of the above set are A = {3,4,5,6}.

**When is Set Builder's method useful?**

Set Builder's method is useful to list any types of sets. However, it becomes really handy when you have to list down a lot of elements, especially infinite number of elements.

For example: A = {x : x is an integer}

It is not possible to list down all the integers within a single set using listing method. So, we can use set builder's method to list down such larger number of elements.