Set Builder Notation

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What is Set Builder Notation?

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”.


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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.


Define Set Builder Notations

The method of defining a set by describing its properties rather than listing its elements is known as set builder notation.

Forming a set in set-builder notation is also known as set comprehension, set abstraction, and set intention.

The set builder notation includes one or more variables and a rule that defines which elements belong to the set and which elements do not belong to the set. This rule is often represented in the form of predicates. The set rule and variables are separated by a vertical slash “|’ or colon (:). This method is widely used for describing infinite sets.

For example, {y : y > 0} is read as: “the set of all y’s, such that y is greater than 0”.


Set Builder Notation Symbols

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

  • The symbol ∈ “is an element of”.

  • The symbol ∉ “is not an element of”.

  • The symbol W denotes the whole number.

  • The symbol Z denotes integers.

  • The symbol N denotes all natural numbers or all positive integers.

  • The symbol R denotes real numbers or any numbers that are not imaginary.

  • The symbol Q denotes rational numbers or any numbers that can be expressed as a fraction.

The set builder notation examples given below will help you to define set builder notation in the most appropriate way. The different set builder notation examples are as follows:


Set Builder Notation Examples

Example

Set Builder Notation

Read As

Meaning

1.

{y : y > 0}

The set of all y such that y is greater than 0

Any Value greater than 0

2.

{y : y ≠ 15}

The set of all y such that y is any number except 15

Any value except 15

3.

{y : y < 7}

The set of all y such that y is any number less than 7

Any value less than 7

4.

{k ∈ Z: k > 4

The set of all Kin Z, such that K is any number greater than 4.

All integers greater than 4


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}.


Set Builder Form or Rule 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.


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 Examples with Solution

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}

FAQ (Frequently Asked Questions)

Q1. How to Express the Domain of a Function in Set Builder Notation?

Ans. The set builder notation can also be used to represent the domain of a function. For example, the function f(y) = √y has a domain that includes all real numbers greater than or equals to 0, because the square root of negative numbers is not a real number. The domain of f(y) in set builder notation is written as:

{y : y ≥ 0}


If the domain of a function includes all the real numbers, (that is there are no restrictions of y), you can simply write the domain as ' all real numbers' or use the symbol R to represent all real numbers.

Q2. What does Unordered Mean in the Set?

Ans. In Mathematics, sets are not organized in a particular order. For example, the set X = {1, 2, 3, 4} seems to be the set of ordered numbers between 1 to 4, but this set is actually equivalent to the set Y = {4, 2, 3, 1}. The order of elements in a set does not matter. Two sets are said to be an equal set if they include all the elements.

Q3. What is the General Form of Set - Builder Notation?

Ans. The general form of set-builder notation is expressed as:

{formula for elements : restrictions} or {formula for elements | restrictions}

Q4. How to Express Inequalities in Set Builder Notation?

Ans. Inequalities in set-builder notation are expressed as:

{x | x ∈ R, x ≥ 2 and x ≤ 8}

This means that the above set includes all the real numbers between 2 and 8 inclusive.