Roster Notation

In this mini-lesson, we will learn about roster notation by understanding the use of roster notation, different types of numbers used in roster notation, and how to apply them while solving problems. We will also discover interesting facts around them.

We all have a habit of arranging our things in a proper manner.

We don't put clothes in a bookshelf nor do we put shoes in a wardrobe. Everything at home has a specific use.

We arrange things to identify them easily. 

We have a similar concept in mathematics where we arrange a particular set of numbers together.

As the name suggests, the term is called sets. A number of sets can be represented by four notations - one of which is called roster notation. 

In this mini-lesson, we will learn more about roster notations. 

Lesson Plan

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Roster notation is one of the most simple techniques to represent the elements of a set.

The elements can be represented in a row and are easy to read and understand.

For example, the set of the first 20 natural numbers divisible by 5 can be represented in roster notation like:

A = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100}

The dotted line at the end represents the continuation of the elements of the set, it means that the numbers after 25 are not shown but it is the part of the set A.

Set A contains only a single element hence the elements can be represented with curly brackets.

\[\text{A = {10}}\]

Set B contains multiple elements hence the elements can be represented as:

\[\text{B = {3, 7, 4, 5, 8, 11}}\]

The elements of set C can be represented as:

\[\text{C = {x, y, z}}\]

One of the limitations of roster notation is that we cannot represent a large number of data in roster notation.

For example, if we want to represent the first 100 or 200 natural numbers in a set B then it is hard for us to represent this much data in a single row.

This limitation can be overcome by representing data with the help of a dotted line.

Take a set of the first 100 positive odd numbers.

\[\text{B = {1, 3, 5, 7........199}}\]

The dotted line shows that the numbers are part of set B but not written in roster notation.

When we represent large numbers of elements in a set using roster notation we usually write the first few elements and the last element and we separate these elements with a comma.

Let's take a set of all the English alphabets, it can be represented like:

\[\text{C = {a, b, c, d.......z}}\]

If any set has infinite numbers of elements like the set of all the even positive numbers, it can be represented like:

\[\text{D = {2, 4, 6, 8......}}\]

We simply can denote the rest of the numbers with a dotted line since there is no end to positive even numbers we have to keep it like this.

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Example 1

 

 

Express the below two sets X and Y in the roster notations.

Solution

The set X in roster notation can be expressed like:

\[\text{X = {1, 2, 3, 4}}\]

The set Y in roster notation can be expressed like:

\[\text{Y = {D, B, C, A}}\]

\(\therefore \text{X = {1, 2, 3, 4}}\)  \[\text{Y = {D, B, C, A}}\]

Example 2

 

 

Express the set \(\text{A = {x | x = 2n^2 - 1, where n \in \mathbb{N} and n<5\) in roster notation.

Solution

The elements of set A is:

for n = 1
\[\begin{align}2n^2 - 1 &= 2 \times 1^2 - 1 \\[0.2cm]
&= 1\end{align}\]

for n = 2
\[\begin{align}2n^2 - 1 &= 2 \times 2^2 - 1 \\[0.2cm]
&= 7\end{align}\] 

for n = 3
\[\begin{align}2n^2 - 1 &= 2 \times 3^2 - 1 \\[0.2cm]
&= 17\end{align}\]

for n = 4
\[\begin{align}2n^2 - 1 &= 2 \times 4^2 - 1 \\[0.2cm]
&= 31\end{align}\]

for n = 5
\[\begin{align}2n^2 - 1 &= 2 \times 5^2 - 1 \\[0.2cm]
&= 49\end{align}\]

\[\text{A = {1, 7, 17, 31, 49}}\]

\(\therefore \text{A = {1, 7, 17, 31, 49}}\)