Set Builder Notation in Sets with Rules and Representation

In Mathematics, set builder notation is a mathematical notation of describing a set by listing its elements or demonstrating its properties that its members must satisfy.

In set-builder notation, we write sets in the form of

{y | (properties of y)}  OR {y : (properties of y)}

Where properties of y are replaced by the condition that completely describes the elements of the set. The symbol ‘|’ or ‘:’ is used to separate the elements and properties. The symbols  ‘|’ or ‘:’ is read as “ such that” and the complete set is read as “ the set of all elements y” such that (properties of y). Here, we are using the variable ‘y’ to formulate the properties of the elements in the set.

Example:

X = {y: y is a letter in the word dictionary}

We read it as,

“X is the set of all y such that y is a letter in the word dictionary”.

What is Set in Mathematics?

In Mathematics, the set is an unordered group of elements represented by the sequence of elements (separated by commas) between curly braces {" and "}.

For example, {cat, cow, dog} is a set of domestic animals, {1, 3, 5, 7, 9} is a set of odd numbers, {a, b, c, d, e} is a set of alphabets.

Let Us Understand The Set Builder Notations

Set Builder Notations is the method to describe the set while describing the properties and not just listing its elements. When there is set formation in a set builder notation then it is called comprehension, set an intention, and set abstraction.

Set builder notation contains one or two variables and also defines which elements belong to the set and the elements which do not belong to the set. The rule and the variables are separated by slash and colon. This is often used for describing infinite sets.

Let Us Check Out The Symbols Used In Set Builder Notation

There are different symbols used for example for element symbol ∈ is denoted for element,  the symbol ∉ is denoted to show that it is not an element, for the whole number it is  W,  symbol Z denotes integers, symbol N denotes all natural numbers and all the positive integers,  symbol R denotes real numbers, symbol Q denotes rational numbers. 

Set Builder Notation Symbols

The different symbols used to represent set builder notation are as follows:

Set Builder Notation Examples

Representation of Sets Methods

There are two different methods to represent sets. These are:

1. Tabular Form or Roasted Method.

2. Set -Builder Form or Rule Method.

Tabular Form or Roasted Method

In the roaster method, the elements of the set are listed inside the braces {}, and each element is separated by commas. If the element appears more than once in the collection, it can be written only once.

Example,

• The set X of the first five natural numbers is written as X = {1,2,3,4,5}.

• The set A of the letter of the word MUMBAI is written as A = {M, U, B, A, I}.

Note: The elements of the set in the roasted method can be listed in any order. Hence, the set {A,B,C,D} can be written as {B, A, C,D}.

How do we Write A Set in Set Builder Method

If the elements of a set have a common property then they can be defined by describing the property. For example, the elements of the set A = {1,2,3,4,5,6} have a common property, which states that all the elements in the set A are natural numbers less than 7. No other natural numbers retain this property. Hence, we can write the set X as follows:

A = {x : x is a natural number less than 7} which can be read as “ A is the set of elements x such that x is natural numbers less than 7”.

The above set can also be written as A = {x : x N, x < 7}.

We can also write, set A = {the set of all the natural numbers less than 7}.

In this case, the description of the common property of the elements of a set is written inside the braces. This is the simple form of a set-builder form or rule method.

Set Builder Form Examples 

Example1: Set of all even numbers: { x | x is an integer and x is even }

• Meaning: The set contains all integers that are even.

Example 2: Set of all positive integers less than 10: { x | x is a positive integer and x < 10 }

• Result: { 1, 2, 3, 4, 5, 6, 7, 8, 9 }

Example 3: Set of all real numbers greater than or equal to 0: { x ∈ R | x ≥ 0 }

• Meaning: The set contains all real numbers 𝑥 that are greater than or equal to 0.

Example 4: Set of all prime numbers: { x ∈ N | x is prime }

• Meaning: The set contains all natural numbers  𝑥 that are prime (e.g., { 2, 3, 5, 7, ... }).

Why do we Use Set Builder Notation?

If you are thinking why do we use such complicated notation to represent sets?  

Or

What is the importance of using such complicated notation?

Now, you can find the answer to this question.

If you are asked to list a set of integers between 1 and 6, inclusive, then you can simply use a roaster form to write {1, 2, 3, 4, 5, 6}.

But the problem may raise if you will be asked to list the real numbers in the same interval in roaster from.

Using the set-builder notation would be convenient to use in this situation.

Starting with all the real numbers, we can limit them to the interval between 1 and 6 inclusive. Hence, it will be represented as:

{x : x ≥ 1 and x ≤ 6}

Set builder notation is also convenient to represent other algebraic sets. For example,

{y : y = y²}

Set-builder notation is widely used to represent infinite numbers of elements of a set. 

Numbers such as real numbers, integers, natural numbers can be easily represented using the set-builder notation. Also, the set with an interval or equation can be best described by this method.

Set Builder Notation for Domain and Range

Set builder notation is a powerful way to describe the domain and range of a function by specifying the values that the input (domain) or output (range) can take.

1. Domain Examples

The domain of a function consists of all possible input values (usually x) for which the function is defined.

Example : Linear Function f(x)=2x+3

• Domain: All real numbers because the function is defined for all x.

• Set Builder Notation:{ x ∈ R | x is a real number }

2. Range Examples

The range of a function consists of all possible output values (usually f(x) or y) that the function can produce.

Example : Linear Function f(x)=2x+3

• Range: All real numbers because the output can take any real value.

• Set Builder Notation: { y ∈ R | y is a real number }

Set Builder Form Examples with Answers

1. Write the given set in the set-builder notation.

A = {1, 3, 5, 7, 9, 11, 13}

Solution: The given set A=  {1, 3, 5, 7, 9, 11, 13} in the set-builder form can be written as:

{x : x is an odd natural numbers less than 14}.

2. How to write x ≤ 3 or x ≥ 4 in set-builder notation?

Solution: We can write x ≤ 3 or x ≥ 4 in set builder notation as:

{x ∈ R |  x ≤ 3 or x ≥ 4}