Roster Form

The roster notation or roster form is a way of representing sets in which the elements are arranged in a row separated by commas and enclosed within curly brackets.

This is the simplest way of writing sets where the order of elements does not matter. 

Example

If ‘A’ is the set of all whole numbers less than 11, then it is represented in roster form as:

A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Roster Form in Venn Diagram

It is easy to represent a set written in roster form using a Venn diagram. 

If A = {1, 2, 3, 6} and B = {3, 4, 5, 6, 7}, when represented in a Venn diagram, we get

From the above Venn diagram, we can easily obtain the set A ∩ B

Here, 

A ∩ B = {3, 6}

Limitations

Since the roster notation lists the elements within curly brackets and separates them by commas, it is difficult to write a set with many elements.

For example, writing a set A of the first 100 whole numbers is difficult. In such a case, we write the first few elements of the set followed by an ellipsis or three dots (…) and then write the last element of the set. 

A = {0, 1, 2, 3, …, 99}

Again, if we have an infinite set, for example, a set of natural numbers N, it can be written in roster form as:

N = {1, 2, 3, 4, …}

Roster Form to Set Builder Form

When representing a set written in roster form to its corresponding set builder form, we first identify the unique property satisfied by all the set elements and then define the set using a mathematical statement or a condition with a variable.

Let us consider a set in the roster form A = {0, 1, 8, 27, 64, …}

Here, the first few elements of the set A can be expressed as

0 = 0³

1 = 1³

8 = 2³

27 = 3³

64 = 4³, and so on.

By identifying the common property of all elements in set A, we get that ‘All the cubes of the given whole numbers start from 0.’

Thus, in set builder form, the set is written as A = {x³ | x ∈ 𝕎}

Roster Notation vs. Set Builder Notation

Here are the basic differences between the roster and the set builder form:

Roster Form	Set Builder Form
The form where we list the elements within curly brackets and separate them by commas.	The form where we use the unique properties or conditions satisfied by all the elements to define the set.
It provides all elements of the set.	It provides a logical condition, statement, or formula that leads to all set elements.
It is convenient to use for the sets with fewer number of elements.	It is convenient to use for sets with a large or an infinite number of elements.
Easy to read and understand.	Requires good knowledge of mathematical concepts to identify the set elements based on a given condition.
For example, Set A = {2, 3, 4, 5}	For example, Set A = {x | 1 < x < 6 and x ∈ ℕ}

Solved Examples