**1. Definition of a set**

A set is a collection of well defined objects.

Examples:

{1, 3, 5, 7, 9} is a set of positive odd numbers less than 10.

The property that the numbers 1, 3, 5, 7 and 9 are odd is their being well defined.

Africa, Asia, America, Australia, Europe are sets of continents in this world.

All students in a class whose names begin with a consonant.

Continent is the well defined property shared by the lands of the above names.

**Note: **

1. A set is denoted by braces {} within which the members of the set are written.

2. The capital letters are used for denoting sets and small letters for members of the sets.

**2. Members/Elements of a set**

The members of a set are called its elements.

In the set {1, 3, 5, 7, 9}, the numbers 1, 3, 5, 7, 9 are called members of the set or more precisely elements of the set {1, 3, 5, 7, 9}

The symbol ‘**∈**’ denotes ‘belongs to’ or it stands for ‘element of’.

Therefore we write 1 **∈**** **{1, 3, 5, 7, 9}, which is read as

‘1 is an element of (or belongs to) the set {1, 3, 5, 7, 9}. Likewise other elements of the set can be similarly denoted.

The symbol ‘’ denotes “does not belong to” or ‘is not an element of’

So, we can translate 2 {1, 3, 5, 7, 9} as

2 is not an element of the set {1, 3, 5, 7, 9}, or

2 does not belong to the set {1, 3, 5, 7, 9}.

Note:

Let A = {1, 3, 5, 7, 9} and B = {5, 7, 9, 1, 3}.

Now are the two sets A and B different?

No. Why?

Because a set is same irrespective of the order in which the elements are written.

**3. Denotation of a Set/Representation of a set. **

** **There are three ways of denoting a set.

**Descriptive method**

In this method, the set is described in complete words in a sentence.

Example 1:

The set A = {1, 2, 3, 4, 5... 100} can be described as

“A is a set of all the consecutive integers from 1 to 100”.

Note:

The three periods after 5 upto 100 stand for all the integers between the two numbers.

**Roster method**

In this method, the set is represented by writing its elements inside braces

{ }

Example 1:

A = {3, 6, 9, 12, 15}

A is a set of the first five positive multiples of 3.

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

X is a set of the vowels in the English alphabets.

**Set builder form**

In this method of representation, the set is described using the unique property shared by all of the elements of the set.

Consider the set

Example 1:

A = {2, 3, 5, 7}

What is the common property shared by the elements 2, 3, 5 and 7.

They are all prime numbers less than 10.

In set builder form, also called Rule method, we write it as

A = {x/x is a prime number less than 10} expanded as

“A is a set of elements x, such that x is a prime number less than 10”

Example 2:

Consider in Roster form the following set

X = {3, 6, 9, 12, 15, 18}.

In set builder form, it is

X = {x/x = 3n, where n ∈N and n < 5}

n ∈N stands for ‘n belongs to the set of natural numbers less than 5

the formula for set builder form is therefore,

S = {x/x has a property} or S = {x: x has a property}