A set is a collection of any particular items that are considered as one object in some respect. For example, set of even numbers less than 45, etc. A set name is always written in capital and the elements of the set are embraced in curly braces. If we represent the set in any other way too then also we embrace the representation of the set within curly braces. There are three ways of representing sets.

1) The first one is description. It means simply the statement of words that describes about the elements of the set. For example: Set A is the set of all positive even numbers that are less than 100.

2) The second way of representing sets is the roster form. It is simply writing all the elements of the sets within curly braces. For example: A = {2,     4, 6, 8, 10, …,94, 96, 98}

3) The last method is known as set builder notation method.
The common sets that are denoted in standard manner are

N -> the set of natural numbers

Z -> the set of all integers

R -> the set of all real numbers

C -> the set of all complex numbers

In set builder notation, we describe the set in the form of words and variables such that following it one can create the complete set elements.

For example: A = {x | x < 0} or A = {x : x < 0}

‘|’ and ‘:’ can either be used. ‘x’ is the variable used to denote the random element of the set and the expression or sentence or most commonly said information after the separation operand explains what type of element it is. Like in our example, x < 0 which implies the set consists of all negative elements.

It is read as “the set of all ‘x’ such that ‘x’ is less than zero”.

In other words set builder notation is a short-hand method used to write the sets in expression manner. It is most commonly used for sets that are consisting of large number of elements or infinite number of elements.

As the name suggests we can build the elements of set with the information given in the set builder form after ‘:’ or ‘|’.
A set builder notation has three parts

1) The braces

2)
Part before | or :

3)
The part after | or :

The braces are the part that is common in any kind of set notation and is most vital as embracement of braces reflects it to be a set.

The part before the separation operand which can be either ‘|’ or ‘:’ read as “such that” is the random variable or variables that is taken to be define the set type like a number or word anything.

The part after the separation is very important as it conditions the set of its type. Like x > 0, x is irrational, x is only even 2x < 10 etc. This condition can be of any sort may it be an expression a condition in words or others.
It is better to understand set builder notation with example. Here are some examples on set builder notation which will give a better understanding on them.

Example:

Write set builder notation for the given sets. Also, write them in roster form.

a) A set of all the elements which are positive integers and a multiple of 3.

b) A set of all natural prime numbers.

c) Set of all integers except 11.

d) Set of elements which are greater than -8 and less than 50.

Solution:

a) The set of all integers which are positive integers and a multiple of 3 can be represented in set builder form as

    A = {x : x $\epsilon$ I$^{+}$& $\frac{x}{3}$ = 0}

    And in roster form as
 
    A = {3, 6, 9, 12, 15, 18, 21, …}

b) A set of all natural prime numbers can be represented in set builder form as

    B = {y : y $\epsilon$ N & y is prime}

    And in roster form as

    B = {2, 3, 5, 7, 11, 13, 17, 19, …}

c) Set of all integers except 11 is represented in set builder form as

    C = {z : z $\epsilon$ I and z $\neq$ 11}

    And in roster form as

    C = {…, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, …}

d) Set of all integers which are greater than -8 and less than 50 is represented in set builder form as

    D = {a : a $\epsilon$ I and -8 < a < 50}

    And in roster form as

    D = {-7, -6, -5, -4, …, 0, 1, 2, 3, …, 45, 46, 47, 48, 49}