Set Theory is a fundamental concept in Class 11 Mathematics that provides a clear way to define and represent collections of objects. Understanding how to represent sets using different methods is essential for solving mathematical problems efficiently. This section focuses on the representation of sets using roster (tabular) form and set-builder form, along with solved examples and practice exercises to strengthen conceptual clarity and problem-solving skills.
- What is Well Defined Collections ?
- What is SET ?
- Set Theory
- Representation of Sets
- What is Roster or Tabular Form of a SET ?
- Practical Examples of Sets in the Roster Form
- What is Rule Method or Set Builder Form of a SET ?
- Practical Examples of Sets in the Set-Builder Form
- What is the Difference Between Roster Form and Set Builder Form of SET ?
- Solved Examples Based on Sets
- Practice Exercise-1
- Practice Exercise-2
- FAQs – Set Theory (Representation of Sets)
What is Well Defined Collections ?
In our mathematical language, everything in this universe, whether living or non-living, is called an object.
In our daily life, while performing our regular work, we often come across a variety of things that occur in groups e.g.
(i) Army of soldiers
(ii) Team of cricket players
(iii) Group of pretty girls
(iv) Pack of playing cards
(v) Bunch of beautiful flowers
The words used above like Army, Team, Group, Bunch, Pack etc. convey the idea of certain collections.
A given collection of objects is said to be well defined, if we can definitely say whether a given particular object belongs to the collection or not.
Well defined collection of objects : If any given collection of objects is in such a way that it is possible to tell, without any doubt whether a given object belongs to this collection or not, then such a collection of objects is called a well defined collection of objects.
Difference Between Well defined collections and Not Well defined collections
| Not well defined collections | Well defined collections |
| (i) A family of rich persons | (i) A family of persons having more than one million rupees. |
| (ii) A group of tall students | (ii) A group of students, with height 170 cm or more. |
| (iii) A group of numbers | (iii) A group of even natural numbers less than 11. |
| (iv) A class of intelligent students in class XI. | (iv) A class of students, who secured 90% or more marks in class XI exams. |
Some more examples of well defined collections:
(i) Vowels of English alphabet, namely $a, e, i, o, u$.
(ii) Odd natural numbers less than 8 i.e. $1, 3, 5, 7$.
(iii) Prime factors of 60, i.e. $2, 3, 5$.
(iv) The roots of the equation $x^2 – 3x + 2 = 0$, i.e. $1$ and $2$.
What is SET ?
Define SET
A well-defined collection of objects is called a set.
The objects in a set are called its members or elements or points.
We denote sets by capital letters $A$, $B$, $C$, $X$, $Y$, $Z$, etc.
If $a$ is an element of a set $A$, we write,
$$
a \in A
$$
which means that $a$ belongs to $A$ or that $a$ is an element of $A$.
If $a$ does not belong to $A$, we write,
$$
a \notin A
$$
Worked Examples of SET
(i) The collection of all vowels in the English alphabet contains five elements, namely $a$, $e$, $i$, $o$, $u$.
So, this collection is well defined and therefore, it is a set.
(ii) The collection of all odd natural numbers less than 10 contains the numbers $1$, $3$, $5$, $7$, $9$.
So, this collection is well defined and therefore, it is a set.
(iii) The collection of all prime numbers less than 20 contains the numbers $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$.
So, this collection is well defined and therefore, it is a set.
(iv) All possible roots of the quadratic equation $x^2 – x – 6 = 0$ are $-2$ and $3$.
So, the collection of all possible roots of $x^2 – x – 6 = 0$ is well defined and therefore, it is a set.
(v) The collection of all rivers of India, is clearly well defined and therefore, it is a set.
Clearly, river Ganga belongs to this set while river Nile does not belong to it.
(vi) The collection of five most talented writers of India is not a set, since no rule has been given for deciding whether a given writer is talented or not.
(vii) The collection of most dangerous animals of the world is not a set, since no rule has been given for deciding whether a given animal is dangerous or not.
(viii) The collection of five most renowned mathematicians of the world is not a set, since there is no criterion for deciding whether a mathematician is renowned or not.
(ix) The collection of all beautiful girls of India is not a set, since the term ‘beautiful’ is vague and it is not well defined.
Similarly, ‘rich persons’, ‘honest persons’, ‘good players’, ‘old people’, ‘young men’, etc., do not form sets.
However, ‘blind persons’, ‘dumb persons’, ‘illiterate persons’, ‘retired persons’, etc., form sets.
Set Theory
The theory of sets was developed by German Mathematician George Cantor (1845–1918 A.D.)
Definition :
A well defined collection of distinct objects is called a set.
Here, the following points are to be noted.
(i) Objects, elements and members of a set are synonymous words.
(ii) Sets are usually denoted by the capital letters $A, B, C, X, Y, Z$ etc.
(iii) The elements of a set are represented by small letters $a, b, c, x, y, z$ etc.
(a) If $a$ is an element of a set $A$, then we say that $a$ belongs to $A$. The word ‘belongs to’ is denoted by the Greek symbol $\in$.
Thus, in notation form, $a$ belongs to $A$ is written as $a \in A$ and $b$ does not belong to $A$ is written as $b \notin A$.
Similarly,
(i) $5 \in N$, $N$ being a set of natural numbers and $0 \notin N$.
(ii) $25 \in P$, $P$ being a set of perfect square numbers, so $5 \notin P$.
(iii) $2 \in Z$, but $0.5 \notin Z$ ($Z$ being the set of integers).
Representation of Sets
Sets are generally represented by the following two ways :
(1) Roster or Tabular form.
(2) Rule Method or Set builder form.
What is Roster or Tabular Form of a SET ?
In this form, all the elements of a set are listed, the elements being separated by commas and are enclosed within brackets $\left\{ \right\}$.
**Note that the order in which the elements are listed, is immaterial.
e.g.
(i) The set of all natural numbers less than 5 is represented as $\left\{1, 2, 3, 4\right\}$.
(ii) The set of all factors of 50 is $\left\{1, 2, 5, 10, 25, 50\right\}$.
(iii) The set of all distinct letters in the word MATHEMATICS is $\left\{M, A, T, H, E, I, C, S\right\}$.
(iv) The set of prime natural numbers is { 2, 3, 5, 7, …}
Practical Examples of Sets in the Roster Form
Problem. Write each of the following sets in the roster form:
(i) $A =$ set of all factors of $24$.
(ii) $B =$ set of all prime numbers between $50$ and $70$.
(iii) $C =$ set of all integers between $-\dfrac{3}{2}$ and $\dfrac{11}{2}$.
(iv) $D =$ set of all consonants in the English alphabet which precede $k$.
(v) $E =$ set of all letters in the word ‘TRIGONOMETRY’.
(vi) $F =$ set of all months having $30$ days.
Solution :
(i) All factors of $24$ are $1, 2, 3, 4, 6, 8, 12, 24$.
$$
\therefore A = \{1, 2, 3, 4, 6, 8, 12, 24\}.
$$
(ii) All prime numbers between $50$ and $70$ are $53, 59, 61, 67$.
$$
\therefore B = \{53, 59, 61, 67\}.
$$
(iii) All integers between $-\dfrac{3}{2}$ and $\dfrac{11}{2}$ are $-1, 0, 1, 2, 3, 4, 5$.
$$
\therefore C = \{-1, 0, 1, 2, 3, 4, 5\}.
$$
(iv) All consonants preceding $k$ are $b, c, d, f, g, h, j$.
$$
\therefore D = \{b, c, d, f, g, h, j\}.
$$
(v) It may be noted here that the repeated letters are taken only once each.
$$
\therefore E = \{T, R, I, G, O, N, M, E, Y\}.
$$
(vi) We know that the months having $30$ days are April, June, September, November.
$$
\therefore F = \{\text{April, June, September, November}\}.
$$
NOTE
We denote the sets of all natural numbers, all integers, all rational numbers and all real numbers by $N$, $Z$, $Q$ and $R$ respectively.
Note :
- In set notation, order is not important. Hence, the set of all factors of 50 can also be written as $\left\{1, 5, 10, 2, 25, 50\right\}$ instead of $\left\{1, 2, 5, 10, 25, 50\right\}$.
- The elements of a set are generally not repeated in a particular set as in example (ii).
- Dots in example (iv) reveals that the list is endless (continued).
What is Rule Method or Set Builder Form of a SET ?
In this form, all the elements of a set possess a single common property $p(x)$, which is not possessed by any other element outside the set. In such a case, the set is described by $\left\{x : p(x)\text{ holds}\right\}$,
We write, $\{x : x \text{ has properties } P\}$.
We read it as, ‘the set of all those $x$ such that each $x$ satisfies properties $P$’.
e.g., in the set $\left\{a, e, i, o, u\right\}$ all the elements possess a common property, each of them is a vowel of the English alphabet and no other letter possesses this property.
If we denote the set of vowels by $V$, then we write
$$V = \left\{x : x \text{ is a vowel in the English alphabet}\right\}$$
The above description of the set $V$ is read as “The set of all $x$ such that $x$ is a vowel of the English alphabet”.
Some other examples are as follows:
(i) Set of all natural numbers less than 5,
$$A = \left\{x : x \in N, x < 5\right\}$$
(ii) Set of all integers,
$$Z = \left\{x : x \in Z\right\}$$
(iii) Set of months having 31 days.
$$A = \left\{x : x \text{ is the month of a year having 31 days.}\right\}$$
Note: It may be observed that we describe the set by using a symbol $x$ for elements of the set (any other symbol like the letter $y, z$ etc. could also be used in place of $x$.).
Practical Examples of Sets in the Set-Builder Form
Q. Write the set $A = \{1, 2, 3, 4, 5, 6, 7\}$ in the set-builder form.
Solution. Given, $A =$ set of all natural numbers less than $8$.
Thus, in the set-builder form, we write it as
$$
A = \{x : x \in N \text{ and } x < 8\}.
$$
Q. Write the set $B = \{1, 2, 4, 7, 14, 28\}$ in the set-builder form.
Solution. Given, $B =$ set of all factors of $28$.
Thus, in the set-builder form, we write it as
$$
B = \{x : x \in N \text{ and } x \text{ is a factor of } 28\}.
$$
Q. Write the set $C = \{2, 4, 8, 16, 32\}$ in the set-builder form.
Solution. Given,
$$
C = \{2^1, 2^2, 2^3, 2^4, 2^5\}.
$$
Thus, in the set-builder form, we write it as
$$
C = \{x : x = 2^n, \text{ where } n \in N \text{ and } 1 \leq n \leq 5\}.
$$
Q. Write the set $D = \{-6, -4, -2, 0, 2, 4, 6\}$ in the set-builder form.
Solution. Clearly, $D =$ set of even integers from $-6$ to $6$.
Thus, in the set-builder form, we write it as
$$
D = \{x : x = 2n, \text{ where } n \in Z \text{ and } -3 \leq n \leq 3\}.
$$
Q. Write the set $E = \{3, 6, 9, 12, 15, 18\}$ in the set-builder form.
Solution. Clearly,
$$
E = \{3 \times 1, 3 \times 2, 3 \times 3, 3 \times 4, 3 \times 5, 3 \times 6\}.
$$
Thus, in the set-builder form, we write it as
$$
E = \{x : x = 3n, \text{ where } n \in N \text{ and } 1 \leq n \leq 6\}.
$$
Q. Write the set
$$
F = \left\{ \dfrac{1}{2}, \dfrac{2}{3}, \dfrac{3}{4}, \dfrac{4}{5}, \dfrac{5}{6}, \dfrac{6}{7}, \dfrac{7}{8}, \dfrac{8}{9} \right\}
$$
in the set-builder form.
Solution. Clearly, we have
$$
F = \left\{ x : x = \dfrac{n}{n+1}, \text{ where } n \in N \text{ and } 1 \leq n \leq 8 \right\}.
$$
Q. Write the set $G = \{1, 3, 5, 7, 9, 11, \dots\}$ in the set-builder form.
Solution. Clearly, $G =$ set of all odd natural numbers.
Thus, in the set-builder form, we write it as
$$
G = \{x : x \in N \text{ and } x \text{ is odd}\}.
$$
Q. Write the set $H = \{1, 4, 9, 16, 25, 36, \dots\}$ in the set-builder form.
Solution. Clearly, $H$ is the set of the squares of all natural numbers.
So, in the set-builder form, we write it as
$$
H = \{x : x = n^2, \text{ where } n \in N\}.
$$
Q. Match each of the sets on the left in the roster form with the same set on the right given in set-builder form:
(i) $\{23, 29\}$ (a) $\{x : x = 3^n, n \in N \text{ and } 1 \leq n \leq 5\}$
(ii) $\{B, E, T, R\}$ (b) $\{x : x = n^3, n \in N \text{ and } 2 \leq n \leq 6\}$
(iii) $\{3, 9, 27, 81, 243\}$ (c) $\{x : x \text{ is prime}, 20 < x < 30\}$
(iv) $\{8, 27, 64, 125, 216\}$ (d) $\{x : x \text{ is a letter of the word ‘BETTER’}\}$
Solution.
(i) $\{23, 29\} =$ set of prime numbers between $20$ and $30$
$\quad = \{x : x \text{ is prime}, 20 < x < 30\}$.
$\therefore$ (i) $\leftrightarrow$ (c).
(ii) $\{B, E, T, R\} =$ set of letters in the word ‘BETTER’
$\quad = \{x : x \text{ is a letter in the word ‘BETTER’}\}$.
$\therefore$ (ii) $\leftrightarrow$ (d).
(iii) $\{3, 9, 27, 81, 243\} = \{3^1, 3^2, 3^3, 3^4, 3^5\}$
$\quad = \{x : x = 3^n, n \in N \text{ and } 1 \leq n \leq 5\}$.
$\therefore$ (iii) $\leftrightarrow$ (a).
(iv) $\{8, 27, 64, 125, 216\} = \{2^3, 3^3, 4^3, 5^3, 6^3\}$
$\quad = \{x : x = n^3, n \in N \text{ and } 2 \leq n \leq 6\}$.
$\therefore$ (iv) $\leftrightarrow$ (b).
What is the Difference Between Roster Form and Set Builder Form of SET ?
| Statement | Roster form | Set builder form |
| (1) The set of currencies used in U.S.A., England, Japan, Germany and Russia. | {Dollar, Pound, Yen, Euro, Rouble} | {x : x is the currencies used in U.S.A., England, Japan, Germany and Russia.} |
| (2) The set of Capitals of Kerala, Karnataka, Tamilnadu, Andhra Pradesh and Gujarat. | {Thiruvananthapuram, Bengaluru, Chennai, Hyderabad and Gandhi Nagar} | {x : x is the capitals of Kerala, Karnataka, Tamilnadu, Andhra Pradesh and Gujarat.} |
| (3) The set of all distinct letters used in the word Student. | {S, T, U, D, E, N} | {x : x is the distinct letters used in the word STUDENT.} |
| (4) The set of all the states of India beginning with the letter A. | {\text{Andhra Pradesh, Arunachal Pradesh, Assam} | {x : x is the state of India beginning with the letter A.} |
| (5) The set of six presidents of India since 1980. | {Neelam Sanjeeva Reddy, Gyani Zail Singh, Radha Swami Venkat Raman, Dr. Shankar Dayal Sharma, K.R. Narayanan, A.P.J. Abdul Kalam, Pratibha Patil, Pranab Mukherjee} | {x : x is the presidents of India since 1980} |
| (6) The set of all natural numbers between 11 and 15. | {12, 13, 14} | {x : x ∈ N, 11 < x < 15} |
Solved Examples Based on Sets
Example 1. Which of the following collections are sets?
(i) The collection of all months of a year beginning with letter J.
(ii) The collection of most talented writers of India.
(iii) The collection of all natural numbers less than 100.
(iv) A collection of most dangerous animals of the world.
[NCERT Exercise 1.1, Q.1]
Sol.
(i) There are 3 months as January, June and July beginning with letter J. Therefore, it is a well defined collection of months and hence, it is a set.
(ii) The concept of most talented writers of India is vague, since there is no rule given for deciding whether a particular writer is talented or not. Hence, the given collection is not a set.
(iii) The collection of all natural numbers less than 100 i.e., $\left\{1, 2, \dots, 99\right\}$ is well defined. Hence, the given collection is a set.
(iv) The concept of most dangerous animals of the world is vague, as there is no rule for deciding whether an animal is dangerous or not. Hence, the given collection is not a set.
Example 2. Let $A = \left\{1, 2, 3, 4, 5, 6\right\}$. Insert the appropriate symbol $\in$ or $\notin$ in each of the following blank spaces:
(i) $5 \dots A$,
(ii) $8 \dots A$,
(iii) $0 \dots A$,
(iv) $4 \dots A$.
[NCERT Exercise 1.1, Q.2]
Sol.
Given, $A = \left\{1, 2, 3, 4, 5, 6\right\}$.
(i) $5$ is an element of set $A$. $\implies 5 \in A$.
(ii) $8$ is not an element of set $A$. $\implies 8 \notin A$.
(iii) $0$ is not an element of set $A$. $\implies 0 \notin A$.
(iv) $4$ is an element of set $A$. $\implies 4 \in A$.
Example 3. Write the set of all consonants in the English alphabet which precedes $k$.
[NCERT Exercise 1.1 Q. 5 (vi)]
Sol.
The consonants which precede $k$ are:
$b, c, d, f, g, h, j$;
Thus, $A = \left\{b, c, d, f, g, h, j\right\}$ is the set of all consonants in the English alphabet, which precede $k$.
Example 4. Write the set of all Indian Miss World since 1995. What value do you expect?
Sol.
The name of all Indian Miss World since 1995 are:
(i) Aishwarya Rai (ii) Yukta Mookhi (iii) Lara Dutta (iv) Priyanka Chopra
Thus, $A = \left\{\text{Aishwarya Rai, Yukta Mookhi, Lara Dutta, Priyanka Chopra}\right\}$.
Example 5. Write the following sets in Roster form:
(i) The set of all natural numbers ‘$x$’ such that $|x + 9| < 50$.
(ii) The set of all positive integers ‘$x$’ such that $|x – 3| < 8$.
Sol.
(i) $4 + 9 < 50 \implies 4x < 41 \implies x < \dfrac{41}{4} \implies x < 10.25$
Since, $x$ is a natural number, so $x$ can take values $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$.
$\therefore$ Required set $=\left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\right\}$.
Case I. $x – 3 = 0 \implies x = 3$
$\implies x – 3 \geq 0 \quad$ i.e. $\quad |x – 3| = x – 3$
$\therefore |x – 3| < 8 \implies x – 3 < 8 \implies x < 11$
The possible values of $x$ are $3, 4, 5, 6, 7, 8, 9, 10$.
Case II. $x < 3 \implies x – 3 < 0$ i.e. $|x – 3| = -(x – 3)$ $\therefore |x – 3| < 8 \implies -(x – 3) < 8 \implies -x + 3 < 8 \implies -x < 5 \implies x > -5$
The values of $x$ are $-4, -3, -2, -1, 0, 1, 2$.
But we have to consider only positive values of $x$, so we ignore negative integers. Thus possible values of $x$ are $0, 1, 2$.
Given set $=\left\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\right\}$.
Example 6. Describe the following sets in Set builder form:
(i) $B = \left\{-1, 5, 9, 61, 67, 71, 73, 79, 83, 89, 97\right\}$
(ii) $C = \left\{14, 21, 28, 35, 42, \dots, 98\right\}$
(iii) $A = \left\{53, 59, 61, 67, 71, 73, 79, 83, 89, 97\right\}$
Sol.
(i) The given set is $\left\{-1, 5, 9, 61, 67, 71, 73, 79, 83, 89, 97\right\}$. We observe that
$|x| = 1$, where $x$ is an integer.
$\therefore$ Required set $=\left\{x : |x| = 1 \text{ and } x \in Z\right\}$
(ii) The given set is $\left\{14, 21, 28, 35, 42, \dots, 98\right\}$.
We observe that these numbers are all natural numbers multiple of 7 and less than 100.
$\therefore$ Given set $=\left\{x : x = 7n, n \in N \text{ and } 7n < 100\right\}$.
(iii) The given set is $\left\{53, 59, 61, 67, 71, 73, 79, 83, 89, 97\right\}$.
We observe that these numbers are all prime numbers between 50 and 100.
$\therefore$ Given set $=\left\{x : x \text{ is a prime number and } 50 < x < 100\right\}$.
Practice Exercise-1
- Which of the following collections are sets and which are not?
(i) The collection of all months of a year beginning with letter A.
(ii) The collection of all natural numbers less than 50.
(iii) The collection of difficult topics in Mathematics.
(iv) The collection of best actors of Bollywood.
(v) The collection of ministers in central Government of India.
(vi) The collection of all months of a year beginning with letter A.
(vii) The collection of all natural numbers less than 50.
(viii) The collection of difficult topics in Mathematics.
(ix) The collection of best actors of Bollywood.
(x) The collection of all months of a year beginning with letter A.
(xi) The collection of all natural numbers less than 50.
(xii) The collection of difficult topics in Mathematics.
(xiii) The collection of best actors of Bollywood.
(xiv) The collection of all months of a year beginning with letter A.
- If $A = \left\{2, 4, 6, 8, 10, 12, 14\right\}$, then insert the appropriate symbol $\in$ or $\notin$ in each of the following blank spaces:
(i) $1 \dots A$, (ii) $6 \dots A$, (iii) $9 \dots A$, (iv) $14 \dots A$, (v) $7 \dots A$, (vi) $10 \dots A$, (vii) $104 \dots A$, (viii) $120 \dots A$. - Describe the following sets in Roster form:
(i) $A = \left\{x : x \text{ is a letter in the English alphabet which precedes } J\right\}$
(ii) $B = \left\{x : x \text{ is a positive integer and } x^2 < 40\right\}$
(iii) $C = \left\{x : x \text{ is an integer and } -3 \leq x < 7\right\}$
(iv) $D = \left\{x : x \text{ is a two-digit natural number such that sum of its digits is 8}\right\}$
(v) $E =$ The set of letters in the word TRIGONOMETRY.
(vi) $F =$ The set of vowels in the word GOD.
(vii) $G =$ The set of all letters in the word MATHEMATICS.
(viii) $H =$ The set of prime numbers less than 30. - Express the following sets by using the Set builder method:
(i) $A = \left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\right\}$
(ii) $B = \left\{1, 3, 5, 7, 9\right\}$
(iii) $C = \left\{0, 3, 6, 9, 12, \dots\right\}$
(iv) $D = \left\{1, 5, 10, 15, \dots\right\}$
(v) $E = \left\{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}\right\}$
(vi) $F = \left\{5, 10, 15, 20, 25\right\}$
(vii) $G = \left\{1, 8, 27, 64, 125, 216\right\}$ - Describe the following sets:
(i) $A = \left\{x : x \text{ is an integer, } x^2 \leq 4\right\}$
(ii) $B = \left\{x : x \text{ is an integer, } -\frac{1}{2} < x < \frac{9}{2}\right\}$
(iii) $C = \left\{x : x \text{ is an odd natural number less than 10}\right\}$
(iv) $D = \left\{x : x \text{ is a letter of the word KUMKUM} \right\}$
(v) $E = \left\{x : x \text{ is a month of the year not having 31 days}\right\}$
(vi) $F = \left\{x : x \text{ is a vowel in the English alphabet which precedes } q\right\}$
(vii) $G = \left\{x : x \text{ is a natural number, } x^4 < 100\right\}$
(viii) $H = \left\{x : x \text{ is a factor of 25}\right\}$
(ix) $I = \left\{x : x = \frac{1}{2n}, n \in N \text{ and } n < 5\right\}$
(x) J = {x : x ∈ Z, –3<x ≤ 2} - Match each of the sets on the left in the Roster form, with the same set on the right described in the Set builder form :
| (i) $\left\{F, A, I, Z, B, D\right\}$ (ii) $\left\{2, 3\right\}$ (iii) $\left\{1, 3, 5, 7, 9\right\}$ (iv) $\left\{3, -3\right\}$ (v) $\left\{0\right\}$ (vi) $\left\{\text{Mr. Mohan Singh}\right\}$ (vii) $\left\{\text{Mr. Pranav Mukharji}\right\}$ (viii) $\left\{1, -1\right\}$ (ix) $\left\{6, 12, 18, 36\right\}$ (x) $\left\{3, 5, 7, 11, 13, 17, 19\right\}$ | (a) $\left\{x : x \text{ is an integer and } x^2 – 9 = 0\right\}$ (b) $\left\{x : x \text{ is an odd natural number less than 10}\right\}$ (c) $\left\{x : x \text{ is a prime number and a divisor of 6}\right\}$ (d) $\left\{x : x \text{ is a Prime Minister of India}\right\}$ (e) $\left\{x : x \text{ is a letter of the word FAIZABAD}\right\}$ (f) $\left\{x : x + 5 = 5, x \in Z\right\}$ (g) $\left\{x : x + x – 2 = 0\right\}$ (h) $\left\{x : x \text{ is a multiple of 6, } x < 40\right\}$ (i) $\left\{x : x \text{ is a prime and odd number less than 20}\right\}$ (j) $\left\{x : x \text{ is a President of India}\right\}$ |
- Write the set of all even integers whose cube is odd.
- Write the set of all real numbers which can not be written as the quotient of two integers in the Set builder form.
- Write the set $\left\{\frac{1}{2}, \frac{2}{5}, \frac{3}{10}, \frac{4}{17}, \frac{5}{26}, \frac{6}{37}\right\}$ in the Set builder form.
- Write the set $A = \left\{x : x \text{ is an integer and } x^2 < 20\right\}$ in the Roster form.
- Write the set $A = \left\{x : x \text{ is a planet}\right\}$ in the Roster form.
- Write the set $A = \left\{x : x \text{ is an Indian nobel prize winner}\right\}$ in the Roster form.
Answers to Practice Exercise-1
- (i) Set (ii) Not Set (iii) Not Set (iv) Not Set (v) Set (vi) Not Set (vii) Not Set (viii) Not Set (ix) Set (x) Set (xi) Set (xii) Set (xiii) Not Set (xiv) Not Set
- (i) $\notin$, (ii) $\in$, (iii) $\notin$, (iv) $\in$ (v) $\notin$, (vi) $\in$, (vii) $\notin$, (viii) $\notin$
- (i) $\left\{a, b, c, d, e\right\}$, (ii) $\left\{1, 2, 3, 4, 5, 6\right\}$, (iii) $\left\{-3, -2, -1, 0, 1, 2, 3, 4, 5, 6\right\}$, (iv) $\left\{17, 26, 35, 44, 53, 62, 71, 80\right\}$, (v) $\left\{T, R, I, G, O, N, M, E, Y\right\}$ (vi) $\left\{0\right\}$ (vii) $\left\{M, A, T, H, E, I, C, S\right\}$ (viii) $\left\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29\right\}$
- (i) $A = \left\{x : x \text{ is a natural number less than 11}\right\}$ (ii) $B = \left\{x : x \text{ is an odd natural number less than 10}\right\}$ (iii) $C = \left\{x : x = 3n, n \in Z^*\right\}$ (iv) $D = \left\{x : x \text{ is a natural number multiple of 5 or } x = 1\right\}$ (v) $E = \left\{x : x = \frac{n}{n+1}, n \text{ is a natural number and } 1 \leq n \leq 6\right\}$ (vi) $F = \left\{x : x = 5n, \text{ where } n \in N \text{ and } n < 6\right\}$ (vii) $G = \left\{x : x = n^3, \text{ where } n \in N \text{ and } n < 7\right\}$
- (i) $A = \left\{-2, -1, 0, 1, 2\right\}$, (ii) $B = \left\{0, 1, 2, 3, 4\right\}$, (iii) $C = \left\{2, 4, 6, 8, \dots\right\}$, (iv) $D = \left\{K, U, M\right\}$, (v) $E = \left\{\text{February, April, June, September, November}\right\}$, (vi) $F = \left\{a, e, i, o\right\}$ (vii) $G = \left\{1, 2, 3\right\}$ (viii) $H = \left\{1, 5, 25\right\}$ (ix) $I = \left\{\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}\right\}$ (x) $J = \left\{-2, -1, 0, 1, 2\right\}$
- (i) $\leftrightarrow$ (c), (ii) $\leftrightarrow$ (c), (iii) $\leftrightarrow$ (b), (iv) $\leftrightarrow$ (a), (v) $\leftrightarrow$ (f) (vi) $\leftrightarrow$ (d), (vii) $\leftrightarrow$ (j), (viii) $\leftrightarrow$ (g), (ix) $\leftrightarrow$ (h), (x) $\leftrightarrow$ (i)
- $\phi$
- $\left\{x : x \text{ is a real an irrational}\right\}$
- $\left\{x : x = \frac{n}{n^2+1} , n \in N, n \leq 6\right\}$
- $A = \left\{-4, -3, -2, -1, 0, 1, 2, 3, 4\right\}$
- $A = \{\text{Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto}\}$
- $A = \left\{\text{Rabindra Nath Tagore, Mother Teresa, C.V. Raman}\right\}$
Practice Exercise-2
1. Which of the following are sets? Justify your answer.
(i) The collection of all whole numbers less than $10$.
(ii) The collection of good hockey players in India.
(iii) The collection of all questions in this chapter.
(iv) The collection of all difficult chapters in this book.
(v) A collection of Hindi novels written by Munshi Prem Chand.
(vi) A team of $11$ best cricket players of India.
(vii) The collection of all the months of the year whose names begin with the letter $M$.
(viii) The collection of all interesting books.
(ix) The collection of all short boys of your class.
(x) The collection of all those students of your class whose ages exceed $15$ years.
(xi) The collection of all rich persons of Kolkata.
(xii) The collection of all persons of Kolkata whose assessed annual incomes exceed (say) ₹ $20$ lakh in the financial year $2016$–$17$.
(xiii) The collection of all interesting dramas written by Shakespeare.
2. Let $A$ be the set of all even whole numbers less than $10$.
(a) Write $A$ in roster form.
(b) Fill in the blanks with the appropriate symbol $\in$ or $\notin$:
(i) $0 \dots A$ (ii) $10 \dots A$ (iii) $3 \dots A$ (iv) $6 \dots A$
3. Write the following sets in roster form:
(i) $A = \{x : x \text{ is a natural number}, 30 \leq x \leq 36\}$.
(ii) $B = \{x : x \text{ is an integer and } -4 < x < 6\}$.
(iii) $C = \{x : x \text{ is a two-digit number such that the sum of its digits is } 9\}$.
(iv) $D = \{x : x \text{ is an integer}, x^2 \leq 9\}$.
(v) $E = \{x : x \text{ is a prime number, which is a divisor of } 42\}$.
(vi) $F = \{x : x \text{ is a letter in the word ‘MATHEMATICS’}\}$.
(vii) $G = \{x : x \text{ is a prime number and } 80 < x < 100\}$.
(viii) $H = \{x : x \text{ is a perfect square and } x < 50\}$.
(ix) $J = \{x : x \in R \text{ and } x^2 + x – 12 = 0\}$.
(x) $K = \{x : x \in N, x \text{ is a multiple of } 5 \text{ and } x^2 \leq 400\}$.
4. List all the elements of each of the sets given below:
(i) $A = \{x : x = 2n, n \in N \text{ and } n \leq 5\}$.
(ii) $B = \{x : x = 2n + 1, n \in W \text{ and } n \leq 5\}$.
(iii) $C = \{x : x = \dfrac{1}{n}, n \in N \text{ and } 2 \leq n \leq 5\}$.
(iv) $D = \{x : x = n^2, n \in N \text{ and } 2 \leq n \leq 5\}$.
(v) $E = \{x : x \in Z \text{ and } x^2 = x\}$.
(vi) $F = \{x : x \in Z \text{ and } – \dfrac{1}{2} < x < \dfrac{13}{2}\}$.
(vii) $G = \{x : x = \dfrac{1}{2n-1}, n \in N \text{ and } 1 \leq n \leq 5\}$.
(viii) $H = \{x : x \in Z, |x| \leq 2\}$.
5. Write each of the sets given below in set-builder form:
(i) $A = \{1, 4, 9, 16, 25, 36, 49\}$.
(ii) $B = \left\{ \dfrac{1}{2}, \dfrac{5}{10}, \dfrac{17}{26}, \dfrac{37}{50}, \dfrac{7}{1} \right\}$
(iii) C = {53, 59, 61, 67, 71, 73, 79}
(iv) D = {-1, 1}
(v) E = {14, 21, 28, 35, 42,…, 98}
6. Match each of the sets on the left described in roster form with the same set on the right described in the set-builder form:
(i) $\{-5, 5\}$ (a) $\{x : x \in Z \text{ and } x^2 = 25\}$
(ii) $\{1, 2, 3, 6, 9, 18\}$ (b) $\{x : x \in N \text{ and } x^2 = x\}$
(iii) $\{-3, -2, -1, 0, 1, 2, 3\}$ (c) $\{x : x \in Z \text{ and } x^2 = 9\}$
(iv) $\{P, R, I, N, C, A, L\}$ (d) $\{x : x \in N \text{ and } x \text{ is a factor of } 18\}$
(v) $\{1\}$ (e) $\{x : x \text{ is a letter in the word ‘PRINCIPAL’}\}$
Answers to Practice Exercise-2
- (i), (iii), (v), (vii), (x), (xii)
- (a) $A = \{0, 2, 4, 6, 8\}$
(b) (i) $\in$ (ii) $\notin$ (iii) $\notin$ (iv) $\in$
- (i) $\{30, 31, 32, 33, 34, 35\}$
(ii) $\{-3, -2, -1, 0, 1, 2, 3, 4, 5\}$
(iii) $\{18, 81, 27, 72, 36, 63, 45, 54, 90\}$
(iv) $\{-3, -2, -1, 0, 1, 2, 3\}$
(v) $\{2, 3, 7\}$
(vi) $\{M, A, T, H, E, I, C, S\}$
(vii) $\{83, 89, 97\}$
(viii) $\{1, 4, 9, 16, 25, 36, 49\}$
(ix) $\{-4, 3\}$
(x) $\{5, 10, 15\}$
- (i) $\{2, 4, 6, 8, 10\}$
(ii) $\{1, 3, 5, 7, 9\}$
(iii) $\left\{1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \dfrac{1}{5}\right\}$
(iv) $\{4, 9, 16, 25\}$
(v) $\{0, 1\}$
(vi) $\{0, 1, 2, 3, 4, 5, 6\}$
(vii) $\left\{1, \dfrac{1}{3}, \dfrac{1}{5}, \dfrac{1}{7}, \dfrac{1}{9}\right\}$
(viii) $\{-2, -1, 0, 1, 2\}$
- (i) $A = \{x : x = n^2, n \in N \text{ and } 1 \leq n \leq 7\}$
(ii) $B = \{x : x = \dfrac{n}{n^2 + 1}, n \in N \text{ and } 1 \leq n \leq 7\}$
(iii) $C = \{x : x \text{ is prime}, 50 < x < 80\}$
(iv) $D = \{x : x \in Z, x^2 = 1\}$
(v) $E = \{x : x = 7n, n \in N, 2 \leq n \leq 14\}$
- (i) $\leftrightarrow$ (c), (ii) $\leftrightarrow$ (d), (iii) $\leftrightarrow$ (a), (iv) $\leftrightarrow$ (e), (v) $\leftrightarrow$ (b)
HINTS TO SOME SELECTED QUESTIONS
(i) Clearly, all whole numbers less than 10 are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
(ii) Clearly, there is no specific criterion to decide whether a given hockey player of India, is good or not. So, the given collection is not a set.
(iii) The collection of all questions in this chapter is a set, because, if a question is given then we can easily decide whether it is a question of the chapter or not.
(iv) Clearly, the term ‘difficult’ is vague. So, the given collection is not well defined and therefore, it is not a set.
(v) Suppose we are given a collection of Hindi novels written by Munshi Prem Chand. Now, if we take any Hindi novel then we can clearly decide whether it belongs to our collection or not. So, the given collection is a set.
(vi) The term ‘best’ is vague. So, the given collection is not a set.
(vii) The given collection has definite members, namely March and May. So, this collection is a set.
(viii) The term ‘interesting’ is vague. So, the given collection is not a set.
(ix) The term ‘short’ is vague. So, the given collection is not a set.
(x) Clearly, it contains definite members. So, it is a set.
(xi) The term ‘rich’ is vague. So, the given collection is not a set.
(xii) The given collection is clearly well defined. So, it is a set.
(xiii) The term ‘interesting’ is vague. So, the given collection is not a set.
Important Chapter Links
In this section on Set Theory: Representation of Sets, Roster or Tabular Form, Set-Builder Form, you will learn how to express sets in different formats for better understanding and application. Designed for Class 11 Mathematics, this chapter includes detailed explanations, solved examples, and practice exercises to help students master set representation. It also supports preparation for JEE PYQs and IMUCET PYQs, enabling students to build a strong foundation for advanced mathematics topics and JEE competitive exams.
FAQs – Set Theory (Representation of Sets)
Q. What is meant by representation of sets?
Ans. Representation of sets refers to the methods used to describe or define a set, mainly through roster (tabular) form and set-builder form.
Q. What is roster (tabular) form of a set?
Ans. In roster form, all elements of a set are listed inside curly brackets, separated by commas.
Q. What is set-builder form of a set?
Ans. Set-builder form defines a set by stating a rule or condition that all elements of the set satisfy.
Q. When should we use roster form?
Ans. Roster form is useful when the number of elements in a set is small and can be easily listed.
Q. When is set-builder form preferred?
Ans. Set-builder form is preferred for large or infinite sets where listing all elements is not practical.
Q. Can a set be written in both forms?
Ans. Yes, most sets can be represented in both roster and set-builder forms depending on convenience.
Q. What are the advantages of set-builder form?
Ans. It provides a concise and clear way to describe sets, especially those with many or infinite elements.
Q. How do you convert roster form to set-builder form?
Ans. Identify the pattern or property of the elements and express it as a rule using a variable.
Q. What are common mistakes in set representation?
Ans. Common mistakes include missing elements, incorrect notation, and not following the defining rule properly.
Q. Why is learning set representation important?
Ans. It helps in understanding advanced topics like relations, functions, and probability, and improves logical reasoning.