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Roster Form in Set Theory: Notation, Examples, and Limitations

Roster Form is one of the two representations that any set can have, with the other representation being Set-Builder Form. In Roster form, all the elements of the set are listed in a row inside curly brackets. If the set comprises more than one element, a comma is used in roster notation to indicate the separation of every two elements. Since each element is counted separately, the roster form is also known as Enumeration Notation.

This article explores the concept of Roster form and helps you learn about this method of representing sets in Set Theory. In addition to details about Roster Form, we will also cover notation, provide examples, and discuss various applications of Roster Form.

Roster-Form

Table of Content

What is Roster Form in Sets?

When representing sets in the roster form, the items are arranged in a row and enclosed in curly brackets. If the set has more than one element, commas are used to separate each pair of elements. For instance, if A is the set of the first 7 natural numbers. In Roster Form, it can be represented by: A = {1, 2, 3, 4, 5, 6, 7}.

Roster Form is also called Tabular Form, as it lists all the elements of the set. In Roster Form the order of elements doesn’t matter as elements in roster form can be written in any order i.e. they don’t need to be in ascending/descending order.

Note: Elements in the Roster Form should not be repeated in the set; thus, in roster notation, elements are only written once.

Let’s consider an example for better understanding.

Example: Write the following elements in roster form.

Elements: 0, 1, 1, 2, 3, 4, 4, 4, 4, 5, 5, and 5

Solution:

A = {0, 1, 2, 3, 4, 5}

All elements are written in any order and only once.

Roster Notation

Roster notation is a way to list the elements of a set in a line, separated by commas, inside of curly brackets i.e., {element 1, element 2, . . . }

The following is an illustration of a set’s roster form:

Example: Represent the first five natural numbers in roster form.

Solution:

First five natural numbers are 1,2,3,4,5

Roster Form of Set A = {1, 2, 3, 4, 5}

Examples of Roster Form

The following example will help us to understand how to represent any data set in day-to-day life in the Roster Form

{1, 2, 3, 4, 5, 6, 7, 8, 9}

{2, 4, 6, 8, 10, 12, 14, 16, 18}

  • Set of Days in a Week:

{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

{2, 3, 5, 7, 11, 13, 17, 19, 23, 29}

  • Set of Vowels in the English Alphabet:

{A, E, I, O, U}

  • Set of Planets in the Solar System:

{Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}

  • Set of the First Five Positive Integers:

{1, 2, 3, 4, 5}

  • Set of Months in a Year:

{January, February, March, April, May, June, July, August, September, October, November, December}

Limitations of Roster Notation

The inability to represent a significant amount of data in roster form is one of the drawbacks of roster notation. It is challenging for us to express this much data in a single row, for instance, if we want to represent the first 1000 or 2000 natural numbers in set A. Data can be represented using a dotted line to get around this restriction. Consider the first 1000 positive even numbers and use roster notation to represent them that is A = {2,4,6,8,…..1000}

The dotted line indicates that although the numbers are not presented in set roster notation, they are a part of set A. When we use roster form to express a large number of elements in a set, we typically write the first few elements and the last element, separating them with a comma. If we were to create a list of every letter in the English alphabet, it would look like this: A = {a, b, c,…. , z}

If a set, such as the set of all positive odd integers, contains an infinite number of elements, it can be written in roster form as A = {1,3,5,7 ,….}. Since there is no limit to positive odd numbers, we must maintain this arrangement and can simply indicate the remaining numbers with a dotted line.

Roster and Set Builder Form

Another notation known as “set builder form” is also used to represent sets. Instead of mentioning the set of all items, we use a condition in this manner to express sets. For instance, the set of vowels in English Alphabets can be expressed as {x | x represents vowels in english alphabets} is the set builder notation. Let’s discuss the difference between both the methods of representation as follows:

Difference between Roster and Set Builder Form

The key differences in both roster and set builder forms are listed in the following table:

Aspect Roster Notation Set Builder Form
Definition Lists the elements of the set explicitly. Describes the set using a rule or condition.
Format Uses braces { } and lists elements separated by commas or semicolons. Uses {x: “condition for x”}
Example {1, 2, 3, 4, 5} {x: x<6, x ∈ N}
Finite or Infinite Can represent finite and infinite sets. Primarily used for infinite sets or sets with a large number of elements.
Readability Easy to read for small sets with a few elements. More concise for describing sets with a pattern or rule.
Common Use Cases Suitable for finite sets or sets with a small number of elements. Used when describing sets with certain properties, e.g., the set of even numbers, prime numbers, etc.
Set Size May not be practical for very large sets. Can describe sets of any size, including infinite sets.
Examples of Infinite Sets {1, 2, 3, . . .} {x: x >0, x ∈ N}

Read More,

Example: Convert the following set from set builder notation into roster notation: P = {x | x is a prime number less than 10}.

Solution:

We know that the prime numbers less than 20 are 2, 3, 5, 7.

Therefore, the given set in roster form is {2, 3, 5, 7}.

Important Points for Roster Form

Let’s summarize the Roster Form in the following important bullets.

  • One way to represent sets is where elements are listed inside curly brackets separated by commas.
  • Elements in Roster Form can be in any order and should not be repeated.
  • It is also called an Enumeration Form.
  • The roster form for an empty or null set is represented by ∅.
  • Roster Form is limited when representing a large amount of data; a dotted line may be used to indicate omitted elements in such cases.

Resources Related to Roster Form

Solved Examples on Roster Form

Problem 1: Find the correct roster form of the set of first three prime numbers from the following:

A = {1, 2, 3}

A = {2, 3, 5}

A = {2, 3, 4}

Solution:

First three prime numbers are 2, 3 and 5.

In roster form, A= {2,3,5}

Problem 2: Write the following sets in roster form.

a. Days in a week

b. First 5 natural numbers

Solution:

Days in a week => A = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

First 5 natural numbers => B = {1,2,3,4,5}

Problem 3: Express the set A = {x | x = 2n2 – 2, where n ∈ N and n < 5} in roster form.

Solution:

The elements of set A are:

For n = 1, 2n2 − 2 = 2 × 12 −2 = 0

For n = 2, 2n2 − 2 = 2 × 22 − 2 = 6

For n = 3, 2n2 − 2 = 2 × 32 − 2 = 16

For n = 4, 2n2 − 2 = 2 × 42 − 2 = 30

For n = 5, 2n2 − 2 = 2 × 52 − 2 = 48

A = {0, 6, 16, 30, 48}

Problem 4: Convert the following set from set builder notation into roster notation: P = {x | x is a prime number less than 15}.

Solution:

We know that the prime numbers less than 15 are 2, 3, 5, 7, 11 and 13.

Therefore, the given set in roster form is {2, 3, 5, 7, 11, 13}.

Practice Problems on Roster Form

Problem 1: Write the following in Roster Form.

  • Set of even numbers between 1 and 10.
  • Set of vowels in the English alphabet.
  • Set of prime numbers less than 20.
  • Set of days in a week.
  • Set of colours in a rainbow.
  • Set of planets in our solar system.

Problem 2: Write the following in Roster Form.

  • Set of three-digit numbers that are divisible by both 3 and 4.
  • Set of all two-digit prime numbers.
  • Set of elements in the periodic table that are noble gases.
  • Set of perfect numbers less than 100.
  • Set of all possible outcomes when rolling a fair six-sided die twice.
  • Set of positive integers that are divisible by 6, 8, and 10.

Roster Form – FAQs

1. Define Roster Form.

Roster form, also known as tabular form, is a method in set theory to list the elements of a set within curly braces, separated by commas or semicolons.

2. Give an Example of Roster Form.

An example of roster form: the set of the first 5 natural numbers divisible by 4 can be represented in roster notation like: A = {4, 8, 12, 16, 20}.

3. What is Roster Notation in Sets?

One of the simplest methods for expressing a set’s constituents is roster notation. The roster notation is a comma-delimited list of elements enclosed in curly brackets.

4. What is the difference between Roster Form and Set Builder Form?

The elements (or members) of a set are listed in a row within curly brackets in the roster form, separated by commas, however in the set-builder form, each element of the set is required to have only one property in order to be a member of the set.

5. How to Express a Set in Roster Form?

In roster notation, if a set contains more than one element, the additional elements are shown in a row enclosed by curly brackets, with commas separating each pair of additional elements.

6. What is the limitation of a set in roster form?

If we have a long list of elements to represent like set of all natural numbers then it is not possible to show all elements individually as a list.

7. What is an Example of a Roster Notation?

An example of roster notation is {1, 3, 5, 7, 9}. This represents a set containing the odd numbers from 1 to 9.


 


Reffered: https://www.geeksforgeeks.org

Class 11

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