Roster and Set Builder Form

A set is a collection of things. There are two methods of representation of sets, namely, roster form and set builder form. Roster form is used to list out the elements of the set while the set builder form describes the elements of the set using a common property. Both the representation of sets have it own advantages and disadvantages. Let's look in the topic of roster and set builder form to know more about the representation of set and its properties.

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This Story also Contains

1. Introduction to Roster and Set Builder Form
2. What is Roster Form?
3. Set Builder Form
4. Comparison between Roster and Set Builder form
5. Solved Examples Based on Roster, and Set Builder form of Sets

In this article, we will cover the concept of the sets and their different forms. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more.

Introduction to Roster and Set Builder Form

A set is a well-defined collection of distinct objects and it is usually denoted by capital letters $\mathrm{A},\mathrm{B}$, $\mathrm{C},\mathrm{S},\mathrm{U},\mathrm{V}$. All the objects that form a set are called its elements or members. These are usually denoted by small letters, i.e. $x,y,z\dots$.
Example: $A=\{1,2,3\}$

There are two methods of representing a set - Roster form and Set-builder Form.

What is Roster Form?

Roster form of set is one way to specify a set, where the elements are listed explicitly within curly brackets. For example, the set of the first five natural numbers is expressed as $\{1,2,3,4,5\}$.

The example of roster form is

$\{\mathrm{a},\mathrm{e},\mathrm{i},\mathrm{o},\mathrm{u}\}$ represents the set of all the vowels in the English alphabet in the roster from.

In roster form, the order in which the elements are listed is immaterial, i.e. the set of all natural numbers which divide 14 is $\{1,2,7,14\}$ can also be represented as $\{1,14,7,2\}$.

An element is not generally repeated in the roster form of a set, i.e., all the elements are taken as distinct. For example, the set of letters forming the word 'SCHOOL' is $\{\mathrm{S},\mathrm{C},\mathrm{H},\mathrm{O},\mathrm{L}\}$ or $\{\mathrm{H}$, $\mathrm{O},\mathrm{L},\mathrm{C},\mathrm{S}\}$. Here, the order of listing elements is not necessary.

Note :

• The elements in roster form can be in any order (they don't need to be in ascending/descending order).
• Same elements should not be repeated in set roster notation.

Roster Notation

In roster form, all the elements of a set are listed, the elements are separated by commas and are enclosed within braces$\{\}$. This notation is straightforward and is generally used when the elements of the set are finite and clear.

Set Builder Form

Here, all the elements of a set possess a single common property that is not possessed by any element outside the set. Let us look on some set builder form examples.

If $Z$ contains all values of $x$ for which the condition $q(x)$ is true, then we write $Z=\{x:q(x)\}\text{ or }Z=\{x\mid q(x)\}$

The $\mathrm{set}A=\{0,1,8,27,64,\dots \}$ can be written in Set Builder form as
$A=\{x^3:X$ is a non-negative integer $\}$

Here, all the elements of a set possess a single common property that is not possessed by any element outside the set.

Write in set builder form.

$Z=\{x:q(x)\}\text{ or }Z=\{x\mid q(x)\}$

The $\mathrm{set}A=\{0,1,8,27,64,\dots \}$ using Set Builder form is $A=\{x^3:X$ is a non-negative integer $\}$

Set Builder Notation

The set builder form of representation of set is enclosed within curly brace $\{\}$. The set builder notation are '$:$ or '$|$' is read as 'such that' respectively.

Conversion Of Roster to Set Builder Form

To convert the roster form to set builder form, first check the similarity among the elements of the given set, then frame a condition based on the similarity.

Example : $A=5,10,15,20,25,30,35,40,45,50$

$A=$$\{x:x$ is a natural number upto $50$ and divisible by $5$}

Conversion Of Set Builder form to Roster Form

To convert the set builder form to roster form, analyse the condition provided and add the relevent element in the set.

Example : $B$ $\{x:x$ is a letter in the word $APPLE\}$

$B=A,P,L,E$

Comparison between Roster and Set Builder form

Recommended Video Based on Roster and Set Builder Form of Sets

Solved Examples Based on Roster, and Set Builder form of Sets

Example 1: Which of the following sets differs from the other three?
1) $A=\{x:x$ is odd, $x\in Z\}$
2) $B=\{x:x$ is not divisible by $2,x\in Z\}$
3) $C=\{x:x$ is the half of an even integer $\}$
4) $D=\{x:x$ is not even, $x\in Z\}$

Solution:
Half of an even integer can be even as well as an odd integer. Eg: half of $4$ is $2$.
All other options denote odd integers.
So, $3$ is different from others.
Hence, the answer is the option 3.

Example 2: Which of the following is a set?
1) The list of all the bright colors.
2) The list of all the dull colors.
3) The list of all colors in the Rainbow.
4) The list of all the good colors.

Solution:
As we learned
$A$ set is a well-defined collection of objects. eg. $A=\{1,2,3\}$.
In this Question,
Bright, dull, and good colors are not well-defined as it is different for different people. But, the list of all colors in the rainbow is definite and well-defined. So, it is a set.

We can decide with respect to any color, say green, whether it will lie in the set or not. So, it is a well-defined collection. We cannot do this in case of bright, dull or good colors.

Hence, the answer is the option 3.

Example 3: Which of these sets are written in the Roster Form?
1) $A=\{x:x$ is a vowel $\}$
2) $B=\{a,e,i,o,u\}$
3) $\mathrm{C}=\{1<\mathrm{x}<2,\mathrm{x}$ is a natural number $\}$
4) $D=\{x:x$ is an even number $\}$

Solution:
As we learned, in roster form, all the elements of a set are listed, and the elements are separated by commas and are enclosed within braces $\{\}$.

Clearly, option 2 is written in roster form.
Hence, the answer is the option 2.

Example 4: Which of the following is not a set?
1) The collection of all licensed drivers in the class.
2) The collection of students in the class above the age $15$.
3) The collection of all the young students in the class.
4) The collection of all students with names starting from ' $A$ '.

Solution:
As we learned
A set is a well-defined collection of objects.
In this question,
"The collection of young students" is not a set because the term young is not well defined.
In all other options, we can identify the elements present in those collections, so they are sets.
Hence, the answer is the option 3.

Example 5: Which of the following sets has an infinite number of elements?
1) $A=\{x:x$ is an odd number on dice $\}$
2) $B=\{x:x$ is a prime number $\}$
3) $\mathrm{C}=\{\mathrm{x}:\mathrm{x}$ is a factor of 24$\}$
4) $D=\{x:x$ is an even prime number $\}$

Solution

Option $1=\{1,3,5\}$ : so finite number of elements.
Option $2=\{2,3,5,7,\dots \}$ : so infinite number of elements.
Option $3=\{1,2,3,4,6,8,12,24\}$ : so finite number of elements.
Option $4=\{2\}$ : so finite number of elements.
Hence, the answer is the option 2.

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