Set Builder Notation

In set builder notation, a set is described by specifying a property that all its elements must fulfill, using a variable (like x or y). 
For example:
Anna is a student who participates in badminton, dance, and quizzes.

Let’s write the set of activities Anna participates in using set builder notation:

A = {a : a is an activity Anna participates in}
This represents the set of her activities.
In words:
A represents the set of all ‘a’, where a is an activity Anna participates in.
 

Set builder notation makes use of various symbols to denote the elements and conditions. Here are a few commonly used symbols:

• The notation ‘|’ or ‘:’ means such that, and it appears after the variable in set builder notation.

•  denotes “belongs to” or “is an element of”.

• ∉ denotes “does not belong to” or “is not an element of”.

• W stands for the set of whole numbers.

• C stands for complex numbers.

• N is the set of positive integers or natural numbers.

• P represents the set of irrational numbers.

• Q represents rational numbers.

• R represents real numbers.
   

There are two different ways to represent a set :

• Roster Form or Tabular Form 

• Set Builder Form or Rule Method

Roster Form or Tabular Form:

A roster form is a list of all the elements of the set enclosed in curly braces {} and separated by commas. This method is also known as the listing method.

In this method, each element can be written only once, even if it frequently appears in the set.

Examples:
The notation for the set of natural numbers from 4 to 8 is expressed as:
X = {4, 5, 6, 7, 8}

The set of letters that make up the word "INDIA" is:
A = {I, N, D, A}

The set {A, B, C, D} can also be written as {B, A, C, D} because the order of the elements is irrelevant.

Set Builder or Rule Method

In the set builder method, a set is defined using a property that is satisfied by all its elements.
Example:
For the set Y = {2, 4, 6, 8}, the set builder notation is:
Y = {x | x is an even natural number less than 10}
This can be read as: “Y represents the set of all elements x such that x is an even natural number less than 10.”
This method is particularly applicable to sets that have numerous elements. It is a more reliable way of representing a set than the roster form. The set builder method is convenient for representing the intervals, conditions, or equations.
 

A set can be described using set-builder notation by defining a property or condition that each of its elements must meet. We define a rule for the elements and utilize a variable (such as x) instead of listing each element.
Standard Format: { x | condition about x }
This means: "The set of all x such that x satisfies the given condition."
The set of all natural numbers smaller than six, for instance, can be expressed as: 
{ x | x ∈ N, x < 6}.
This indicates that x is a natural number and x is less than 6.
 

Set-builder notation provides a condition that applies to each element in a set. To read it correctly, we need to understand its structure and components. 
It is written in the form: 
B = {x | condition about x}, 
This means: “B represents the set of all x such that the given condition about x is true.”
Write it using a colon:
B = {x: condition about x}
We can use the “|” symbol instead of “:” since both imply “such that”.
 

You might think about the actual purpose of set builder notation. Here is the answer to your question. The roster form can be used for smaller sets of integers, but set builder notation is more reliable for larger sets of elements.
For example, it is not possible to express all the real numbers in the range of 1 to 6 in roster form.
Instead, we can use set builder notation:
{x ∈ ℝ : 1 ≤ x ≤ 6 }
This denotes: “x is a real number such that x lies between 1 and 6 inclusive.”
 

When writing the domain and range of a function, we use the set builder notation. The domain of a function includes all values entered into it. For instance, all real numbers other than 1 would fall within the domain of the rational function f(x) = 2/(x-1). When x = 1, the function f(x) would not exist.  Thus, {x ∈ R | x ≠ 1} is the expression for the domain of this function.
Set builder notation can also be used to express a function's range.  All possible output values for the function are included in the range.

For the function f(x) = 2 / (x - 1), we write the following:
y = 2 / (x - 1)
Next, solve for x in terms of y
x - 1 = 2 / y
x = (2 / y) + 1.

We observe that there is a matching value of x for each value of y ≠ 0, indicating that all such y-values are attainable.

But when y = 0, the equation changes to:

The value x − 1 = 2 / 0 is not specified.  Thus, y = 0 is outside the range.

Therefore, the range includes all real values other than 0.
In the set builder notation, the range is thus written as follows: {y ∈ R | y ≠ 0}.