Roster Form and Set Builder Form: Definition, Cardinality, Examples

Roster Form and Set Builder Form: We see many items and things around us, some individually and others in groups. An example of a collection of objects is a school of fish, a swarm of bees, a pride of lions, keys, etc. In the same way, a set is a well-defined collection of objects, numbers, or things. There are various ways to represent a set. A set can be expressed in three ways, such as descriptive, roster, and set-builder.

Listing the elements of a set inside a pair of braces { } is called the Roster Form. On the other hand, in Set Builder Form, the statement is enclosed within brackets, which allows for a better definition of the set. All elements of a set must possess the same property in the Set Builder form to become a member of that set. Scroll down to know more about it!

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Define Set

A set is a collection of well-defined objects. Well-defined means it must be obvious which things belong to the set and which do not. For example, a group of beautiful flowers is not a set because the flowers to be included in the set are not well-defined. The word beautiful is a relative term. What may be beautiful to one person may not be the same for the other person, and thus, the collection of lovely flowers cannot form a set. On the contrary, if we say a collection of yellow colour flowers, it will create a set because every yellow flower will be included.

Elements of a Set

The objects used to form a set are called its elements or its members. In general, the elements of a set are written inside the curly braces and separated by commas. The name of the set is always written in capital letters.

For example,
\(A =\){Messi, Pique, Griezmann, Roberto, Busquets}
Here \(A\) is the set’s name, whose elements or members are Messi, Pique, Griezmann, Roberto, and Busquets.
The Greek letter epsilon \(ϵ\) is used for the words belongs to, is an element of, etc. Thus, \(x ϵA\) will be read as \(x\) belongs to set \(A\) or as \(x\) is the element of the set \(A\). The symbol \(∉\) stands for does not belongs to or is not an element of the set.

Properties of Sets

Given is the list of some properties of sets. Let us have a look at it.

1. The elements of a set are written inside a pair of curly braces and separated by commas.
2. The set is represented by a capital letter.
3. If the elements of a set are alphabets, these elements are written in small letters.
4. The elements of a set can be written in any order.
5. No elements in a set must be repeated.

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Representation of Sets

For representing a set, the following given methods are commonly used.

1. Description Method
2. Roster or Tabular Method
3. Rule or Set-Builder Method

Description Method

In this method, a well-defined description of the elements of a set is made. At times, the definition of elements is enclosed within the curly brackets.

For example, a set of football players with ages between \(20\) years and \(30\) years can be written in the description method as
\(A =\) {football players with ages between \(20\) years and \(30\) years}

Roster or Tabular Method

In the roster method, the set elements under consideration are written inside a pair of curly braces and are separated by commas. For example, if \(A\) is the set of the first \(5\) months of the year, then
\(A =\) {January, February, March, April, May}
If \(X\) is the set of whole numbers less than \(10\), then
\(X = \{ 0,\,1,\,2,\,3,\,4,\,5,\,6,\,7,\,8,\,9\} \)

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Rule or Set Builder Method

In this method, the actual elements of the set are not listed; relatively, a brief or a statement or formula is written inside a pair of curly braces. For example, if \(P\) is the set of counting numbers greater than \(10\), then set \(P\) in set-builder form can be written as
\(P =\) {\(x:x\) is a counting number greater than \(10\)}
or it can be written as
\(P =\) {\(x|x\) is a counting number greater than \(10\)}

This will be read as \(P\) is a set of elements \(x\) such that \(x\) is a counting number greater than \(10\).
The symbols’\(:\)’ and ‘\(|\)’ stand for such that.

Roster Form to Set-Builder Form

Let us understand the conversion of the roster form to the set-builder form with the help of examples.
Example 1: \(A = \{ 5,\,10,\,15,\,20,\,25,\,30,\,35,\,40,\,45,\,50\} \)
Solution: \(A =\) {\(x:x\) is a natural number up to \(40\) and divisible by \(5\)}

Example 2: \(B = \{ 1,\,8,\,27,\,64,\,125\} \)
Solution: \(B =\) {\(x:x\) is a perfect cube natural number up to \(125\)}

Example 3: \(C = \{a,\,e,\,i,\,o,\,u\}\)
Solution: \(C =\) {\(x:x\) is a vowel in English alphabet}

Observing the relationships between the set elements and writing the condition as a statement to change from roster form to set builder form.

Set-Builder Form to Roster Form

Let us understand the conversion of set-builder form to roster form with the help of examples.

Example 1: \(A =\){\(x:x\) is a letter in the word \(BARCELONA\)}
Solution: \(A = \{b,\,a,\,r,\,c,\,e,\,l,\,o,\,n\}\)

Example 2: \(Q =\){\(x:x\) is an even number between \(9\) and \(19\)}
Solution: \(Q = \{10,\,12,\,14,\,16,\,18\}\)

Example 3: \(T =\){\(x:x\) is a month of a year not having \(30\) days}
Solution: \(T=\){January, February, March, May, July, August, October, December}

Example 4: \(B =\){\(x:x\) is an integer lying between \(-5\) and \(3\)}
Solution: \(B = \{-4,\,-3,\,-2,\,-1,\,0,\,1,\,2\}\)

To convert from set-builder form to roster form, arrange and write the objects in the set as per the given conditions in set-builder form.

Cardinality of a Set

The number of elements in a set is called its cardinality. For example, \(A = \{m,\,e,\,r,\,u,\,t\}\) has five elements. Thus, the cardinal number of set \(A = 5\).
\(Q={}\) has no elements. Therefore, the cardinal number of set \(Q = 0\).
The symbol representing the cardinal number is the small letter \(n\) attached before the set’s name, written inside brackets. Thus the cardinal number of set \(A\) is represented by \(n(A)\).

On the contrary, \(n(P)\) represents the cardinal number of set \(P\).
Let us understand it with the help of an example.

Example 1: Write the cardinal number of each of the following sets.
\(P =\){counting numbers between \(5\) and \(30\), that are divisible by \(5\)}
Solution: Since \(P = \{10,\,15,\,20,\,25\}\)
Therefore, \(n (P) = 4\)

Solved Examples

Q.1. Write each of the following sets in the roster form and set-builder form.
a) Set of all two-digit numbers that are perfect square
b) Set of all natural numbers that can divide 24 completely.
Ans: a) In the roster form: \(A = \{16,\,25,\,36,\,49,\,64,\,81\}\)
In the set-builder form: \(A =\) {\(x:x\) is a two-digit perfect square number}
b) In the roster form: \(B = \{1,\,2,\,3,\,4,\,6,\,8,\,12,\,24\}\)
In the set-builder form: \(B =\) {\(x:x\) is a natural number which divides \(24\) completely}

Q.2. Write each given set in the roster form.
a) Five cities of India whose name start with the letter D
b) Any five geometrical figures
Ans: a) \(A =\){Dehradun, Delhi, Dhanbad, Dharamsthala, Dindigul}
b) \(B =\){Triangle, Rectangle, Kite, Hexagon, Square}

Q.3. Write each given set in the set-builder form.
a) \(P = \{2,\,4,\,6,\,8,\,10\}\)
b) \(Q =\){FC Barcelona, Real Madrid, Atletico Madrid, Sevilla FC, Valencia CF}
Ans: a) \(P =\){\(x:x\) is an even natural number less than \(12\)}
b) \(Q =\) {\(x:x\) is a name of top five football team of La Liga Spanish league}

Q.4. Convert the following given in roster form to set-builder form.
a) \(A = \{1,\,3,\,5,\,7,\,9,\,11,\,13,\,15\}\)
b) \(P =\) {Tuesday, Thursday}
Ans: a) \(A =\){\(x:x\) is an odd natural number less than \(17\)}
b) \(P =\) {\(x:x\) is a day of the week whose name starts with \(T\)}

Q.5. Convert the following given in the set-builder form to the roster form.
a) \(H =\) {\(x:x\) is a two-digit natural number which is a multiple of \(11\)}
b) \(Q =\) {\(x:x\) is a natural number in between \(99\) and \(110\)}
Ans: a) \(H = \{11,\,22,\,33,\,44,\,55,\,66,\,77,\,88,\,99\}\)
b) \(Q = \{100,\,101,\,102,\,103,\,104,\,105,\,106,\,107,\,108,\,109\}\)

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Summary

In this article, we learned about sets, properties of sets, and elements of a set. Then we learned about the three methods to represent a set- Description Method, Roster or Tabular Method, and Rule or Set-Builder Method. In addition to this, we learned to convert the roster form to set-builder form and vice versa. Furthermore, we learnt the cardinality of a set.

Frequently Asked Questions (FAQs) – Roster Form and Set Builder Form

Let’s look at some of the commonly asked questions about the Roster Form and Set Builder Form in the following section:

Q.1. What is the difference between roster form and set builder form?
Ans: In the roster form, the listed elements are written inside a pair of curly braces and are separated by commas, whereas in set-builder form, a brief or a statement or formula is written inside a pair of curly braces.

Q.2. What is set builder form? Give an example.
Ans: In set-builder form, the actual elements of the set are not listed; relatively, a brief or a statement or formula is written inside a pair of curly braces. For example, if \(P\) is the set of counting numbers greater than \(20\), then set \(P\) in set-builder form can be written as
\(P =\) {\(x:x\) is a counting number greater than \(20\)}

Q.3. Give an example of the roster method?
Ans: \(=\){ \(x:x\) is a month of a year having \(30\) days}
\(T=\){April, June, September, November}

Q.4. How do I convert a roster form to a set-builder?
Ans: Let us understand the conversion of the roster form to a set-builder form.
If a set \(A\) is written in the roster form, like, \(A =\){January, June, July}, then in set-builder form, it can be written as \(A=\){\(x:x\) is a month whose name starts with \(J\)}

Q.5. How do you write a set in roster form?
Ans: In the roster method, the set elements under consideration are written inside a pair of curly braces separated by commas. If \(A\) is a set of integers in between \(- 10\) and \(10\), then in roster form, it can be written as;
\(A = \{- 9,\,- 8,\,- 7,\,- 6,\,- 5,\,- 4,\,- 3,\,- 2,\,- 1,\,0,\,1,\,2,\,3,\,4,\,5,\,6,\,7,\,8,\,9\}\)

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