Set Builder Notation: Explained In The Simplest Manner

Set Builder Notation: Explained In The Simplest Manner

The term “set builder” is defined as a mathematical term used to define a set with symbols. It’s used to describe the elements of sets, their relations, and the operations between the sets. A set of numbers, distinct components are referred to as a set. We’ll also see some set builder notation examples here.

What Is a Set?

There are objects everywhere around us. Certain of them are put together, and others are an array of objects. A set is a collection of things, objects or numbers.

If we can determine and identify the components of a particular collection, that collection is known as a set. The most common way to use capital letters is to refer to the set as A, B,….., and so on. Also, The content or elements that comprise the entire set can be identified with lower case letters like A, B, C,…, and the like.

Methods of Writing Set

Two ways can be employed to depict the set-

(1) Method of listing or method of roster

In this way, we take note of all the components of a set within curly brackets.

Each element is only written once and divided by commas.

For instance, the sequence of letters that make up the word “California” is written as A = c, G, P, O, r.

(2) Rules method, or set the form of a builder

In this way, it isn’t mandatory to write down the elements. Instead, we write the representative element using the variable followed by an arrow or a colon and then add the basic property for that representative element.

Sets and Roster Form

A set is an assortment of different objects, and they are referred to like parts in the set. They can include anything from numbers, forms, colours, symbols, animals, and other sets. 

There are also sets of numbers. Take the example of a set of even numbers in Between 10 and 20 inclusive.

It is possible to represent the set by drawing a picture. However, we could also depict this set using a completely different manner in the form of a roster.

To create a set in the form of a roster, you only have to list every component of the set separated by commas inside the curly braces!

The set-builder note is composed of three parts.

  • Variable
  • Separator
  • Formula

Let’s go over each element of the set-builder format easily.

A variable: A variable can represent every member (or element) of the set.

A separator: A vertical bar or colon is used to differentiate between the formula and the variables in the notation. The separator can be understood in the form of “such that” or “with the property that” or “for which.”

A rule: Also, The term “set” is defined by looking at one or more common features in the element. It is usually expressed as the logic formula or a predicate (or formula).

Sets: They are among the most effective ways to visualize connections between various classes of data. The concept of sets can help us deal with various connections in Differential Geometry, fields of functions, and various other fields.

To perform these tasks, sets require specific symbols to ensure uniform representation. Once we are familiar with these notations, it’ll become easier to comprehend sets and their relations. These notations allow us to analyze set relations universally or uniformly and also make the representations simpler.

What Is Set Builder Notation?

Let’s describe Set Builder Notation and Interval Notation here.

Set builder notation used in sets represents elements by illustrating their relationships instead of showing the entire set of elements.

For example, if I wish to represent an even natural set of numbers less than 100, it will take a long time to record every component in a roster format, so this is set builder format. It could be represented as :

A=x, x = 100

Interval notation allows you to specify a set of numbers that fall between the lower limit and the upper limit, using the values of an endpoint.

Lower and upper limits could or might be included in a set.

The values at the end of the line are written in parentheses or brackets.

A square bracket signifies being included in the group, while a parenthesis signifies excluding the set.

For instance,


The interval notation above indicates that this set comprises the real numbers between 8 and 16. 16 is part of the set, while 8 isn’t a component of this set.

Interval Notation:

Interval notation is yet another way of describing and specifying the sets, which includes each of the numbers in between the lower limit, which could be or not be included, and an upper limit that could be included or not.

A square bracket signifies it is part of the interval. A round bracket signifies that the limit isn’t part of the interval. The way to indicate this is as follows:

Interval : (-∞ , 10 ]

Why Use Set Builder Notation?

It is possible to ask what’s the need for this complex notation! What’s the purpose of the set-builder notation?

If you need to write an integer range in the range 12-16, One can utilise the roster notation to write 12,13,14,15,16.

The problem comes when we need to write the actual numbers within the same time frame.

The use of roster notation wouldn’t be feasible.

{, 1.1, 1.01, 1.001, 1.0001, .. ??? }

The set-builder notation could be more appropriate in this case.

Numbers such as real, natural, and integers can be conveniently expressed using this set-builder notation.

Using this method, we can also express An interval set or equation.

How to use a Set Builder Notation?

The Set Builder Notation is a math form of notation that describes the set listing various properties that elements of the set must meet. It can be beneficial to explain sets with a never-ending number of elements.


In conclusion, we will present a piece of knowledge that is vital-

Set builder notes have three major components:

1. We generally write a variable is in lowercase

2. Vertical bar separator, also we call a colon, can be taken to mean “such that”

3. Logical phrases define the characteristics of sets.

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