An Overview of Set Builder Notations, its Uses, Examples, and Solved Problems

There are so many concepts in mathematics that we can use to make our life easygoing, and sets are one of them. Sets are a part of mathematics that help group data within two curly braces without entering the data. Sounds confusing? No worries.

In this section, we will be explaining sets, set builder notation symbols, and their uses. For better understanding, we have added some examples and solved questions to practice with them. 

Introduction to Sets

A collection of elements presented inside curly braces with the help of commas are known as sets. Sets help in grouping numbers of various objects termed as elements. 

Sets are usually denoted with capital letters of the English alphabet, like X, Y, Z, A, B, C, etc. In contrast, elements are presented as small letters, like a, b, c…x,y,z, and so on. 

Let’s understand with the help of some examples, {Mango, Banana, Orange, Grapes} is forming a set of fruits, {2,4,6,8} is forming a set of first four positive even numbers (Even numbers are those numbers that are divisible by 2). 

Representation of sets

The sets can be represented with the help of two different methods, as follows:

  1. Roster/ Tabular notations 
  2. Set- builder notations

Let’s have an overview of the two:

  • Roster notations

This method is a very basic way of forming sets. Elements need to be added to the sets as they are while using this method.

The elements in the roster method are separated with the help of commas.  For example, if we have a set X of the first 6 natural numbers, then the set will be as follows;

X= { 1,2,3,4,5,6 }

One element cannot be written twice in roster notations; however, they can be arranged in any order. 

  • Set builder Notations

Set builder notations are those mathematical notations that help represent a set of elements with the same characteristics together without even actually representing them within curly braces. 

Elements of sets will depend on their properties; if they have similar properties, only the elements will be placed in the same set. 

Formation of sets in set builder notations is also termed set comprehension, set abstraction, and set an intention. 

The general form of writing sets using set builder notations is:

“{a|( Properties of a)}” or “{a: ( Properties of a)}”, here, “a” represents the elements, “|” or “:” separates the elements and its properties, these symbols are also termed as “such that”.  The above set will be read as ” the set of all elements a” such that ( properties of a). Here, “a” is the variable. 

Let’s take an example:

X= {p: p is a vowel from English alphabets } 

We will read the above set as: 

“X is the set of all p such that p is a vowel from English alphabets”.  

In set-builder notations, sets can have more than one variable; those sets have a rule that addresses which elements belong to the set and which do not. The rule is usually represented as predicates. 

Symbols that we have discussed before, “:” or ”|” come into use while separating rules from properties. 

For example: 

X= { p : p < 0 } will be read as “the set of all p’s, such that p is smaller than 0”. 

Set builder notation symbols 

There is a limited number of symbols present in set builder notations. These are as follows:

  • The symbol ∈ denotes “belong to.” 
  • The symbol denotes “doesn’t belong to”
  • The symbol N denotes “all natural numbers or all positive integers.”
  • The symbol W denotes ” whole numbers.”
  • The symbol Z denotes “integers.”
  • The symbol R denotes “real numbers.”
  • The symbol Q denotes “rational numbers.”

Read the examples given below to clear your doubts.  The examples are:

  1. { p: p > 10 }, this set builder notation will be read as the set of all p such that p is greater than 10. 

Result:  Any value greater than 10. 

  1. { p : p < 0 }, this set builder notation will be read as the set of all p such that p is smaller than 0. 

Result: Any value greater than 0

  1. A = { y: y ∈ N, 1< y <10 }, this set builder notation will be read as the set of all y that belongs to natural numbers and lie between 1 to 10. 

Result: Any number that lies between 1 to 10. 

Uses of set-builder notation 

Set builder notations are the ones that usually come in use at the time of building sets. But why do they? 

Let’s discuss some of the best uses of set builder notations:

  • Set-builder notations make it easy to form sets of a large group of numbers that can’t be formed using roster notations. For example, if you are asked to form a set of the first 5 natural numbers, you can do it easily. The set will be in { 1,2,3,4,5 }.

However, if you are asked to form a set of all the natural numbers, then it’s impossible to add all the natural numbers in roster form. Using set-builder notations, the set will be, { y: y is a natural number } 

  • Set-builder notations can also be used to represent other algebraic sets. Let’s understand with the help of an example, { a: a= a³ } 
  • Set-builder notations can form sets using any kind of values: real values, natural values, whole numbers, etc. 
  • This method can make it simple to form sets that consist of intervals. 

Solved problems on set-builder notations

  1. Write a set of natural numbers more than 1 but less than 10 using set-builder notation.

Solution: X= { y: y ∈ N, 1 < y < 10 } 

  1. Write the given set in set-builder form:

{ 2,4,6,8,10 }

Solution: X= { y: y is an even number, y< 1<11} 


The set-builder notations are widely used because of their nature of solving difficult problems related to sets in a matter of time. 

We can call set-builder notations a shorthand use of building sets. It helps in making sets using any kind of numbers irrespective of their nature and their number. We hope this article will help you revise the set-builder notation when you have your exams right around the corner. 

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