Sets (Maths) - Definition, Types, Symbols & Examples

Definition of Sets

A set is a collection of well-defined objects. The objects which are in the set are called the elements of a set.

Example: Let us consider $A$ as the set of all natural numbers till $7$. So, set $A = \{1,2,3,4,5,6,7\}$.

Elements of Set

The objects which are in the set are called the elements of a set.

Example: Let us consider $A$ as the set of all natural numbers till $7$. So, set $A = \{1,2,3,4,5,6,7\}$. Here $1,2,3,4,5,6,7$ are the elements of set $A$.

If $a$ is an element of set $A$, it can be said “$a$ belongs to $A$” and represented using $\in$ in place of ‘belongs to’. It can be represented as $a \in A$.

Order of Sets

Order of set is also known as the cardinality of set. The number of elements in a set is called its cardinal number or cardinality of a set. It is denoted by $n(A)$. If $\mathrm{A}=\{\mathrm{a}, \mathrm{s}, \mathrm{d}\}$, then $n(A)=3$

and if $B=\left\{x: x^2=1\right\}$, then $B=\{1,-1\}$, and hence $n(B)=2$

Set Symbols

Representation of Sets

Sets can be represented in two ways, namely Roster and Set Builder Form.

Roster Form(Tabular form)

Roster form is one of the ways to represent a set. In this form, the elements of the set are listed implicitly within curly brackets($\{\}$).

Example: ${a,e,i,o,u}$ is the set of all vowels in English alphabets.

The elements in roster form can be in any order (they don't need to be in ascending/descending order). 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 $\{S,C,H,O,L\}$ or $\{O,H,L,S,C\}$.

Set-Builder Form

In set builder form, the set is defined using the common property of the elements. For example, If $Z$ contains all values of $x$ for which the condition $q(x)$ is true, then we write

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

Where, ': ' or '|' is read as 'such that'.

Difference between Roster and Set-Builder Form

Types of Set

The types of sets in mathematics are as follows,

1. Empty Set: A set that does not contain any element is called an empty set, void set or null set. Eg. The set of mangoes in the basket of guavas.

2. Singleton Set: A set having only one element is called a singleton set. Eg. Set of all whole numbers which is not a natural number which is $A = \{0\}$

3. Finite Set: An empty set or a set consisting of a finite number of elements of a set is called finite set. Eg. Set of all natural numbers less than $7$ that is $A = \{1,2,3,4,5,6\}$

4. Infinite Set: A set consisting of an infinite number of elements of a set is called an infinite set. Eg. Set of all whole numbers which is $A=\{1,2,3,4,5,6,7,....\}$

5. Equivalent Set: Two sets having the same number of elements are called equivalent sets. For sets $A$ and $B$, it is represented as $n(A) = n(B)$.

e. g. Let $A$ be the set of all natural numbers less than $6$ and $B$ be the set of all whole numbers less than $5$.

$A=\{1,2,3,4,5\}$

$B=\{0,1,2,3,4\}$

Here, $n(A) = 5$ and $n(B) = 5$

Therefore, the sets $A$ and $B$ are equivalent sets.

6. Equal Set: Two sets having the exact elements in both sets are called equal sets. For sets $A$ and $B$, it is represented as $A$ = $B$.

Eg. Let $A$ be the set of all natural numbers less than $6$ and $B$ be the set of all whole numbers greater than $0$ and less than $6$.

$A=\{1,2,3,4,5\}$

$B=\{1,2,3,4,5\}$

Here, $A = B$

Therefore, the sets $A$ and $B$ are equal sets.

7. Disjoint set: Two sets $A$ and $B$ are disjoint if they do not have any common element. That is, $A$ and $B$ are disjoint if $A \cap B=\emptyset$. (i.e) Two sets are said to be disjoint if each set has distinct elements in it.

e.g. Let $A$ be the set of all natural numbers and $B$ be the set of all integers less than $0$.

$A = \{1,2,3,4,5,6,....\}$

$B = \{-1,-2-3,-4,-5,-6,.....\}$

Here, there is no common element in the set $A$ and $B$. So, the sets $A$ and $B$ are disjoint sets.

8. Power Set: The set of all possible subsets of a set is called a power set. The power set always contains $2^n$ elements, where $n$ is the number of elements in the original set.

e.g. Let $A = {1,2,3}$

The number of elements in the power set is $2^n = 2^3 = 8.$

The power set of $A$, $P(A)$ is $\{\phi, \{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{1,3\},\{1,2,3\}\}$

9. Universal Set: A set that contains all related sets in a given context is called the Universal Set. The universal set is usually denoted by $U$ while all its subsets are denoted by the letters $\mathrm{A}, \mathrm{B}, \mathrm{C}$, etc.

e.g. Let $A$ be the set of all natural numbers and $B$ be the set of integers greater than $-10$ and less than $5$.

$A = \{1,2,3,4,5,6,....\}$
$B = \{-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4\}$

The universal Set $U$ may be $\{-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,...\}$ or the set of all integers($Z$) or any larger set containing all the elements of the subsets.

Operations on Sets

The operations valid on sets are,

1. Union: Union of two sets combines the values in both sets without repetition to form a set. The symbol ' $\cup$ ' is used to denote the union.

Eg. $A = \{1,2,4,5,7,8,9\} , B = \{9,7,21,34\}$

Then union of sets $A$ and $B$, $A \cup B = \{1,2,4,5,8,9,7,21,34\}$

2. Intersection: Intersection of two sets is a set containing the common elements on both the sets. The symbol ' $\cap U$ ' is used to denote the intersection.

Eg. $A = \{1,2,4,5,7,8,9\} , B = \{9,7,21,34\}$

Then intersection of sets $A$ and $B$, $A \cap U B = \{7,9\}$

3. Complement: Let $U$ be the universal set and $A$ is a subset of $U$. Then the complement of $A$ is the set of all elements of $U$ which are not the elements of $A$. Symbolically, we use $A^{\prime}$ or $A^C$ to denote the complement of $A$ with respect to $U$.

Eg. Let the universal Set $U$ be $\{-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,...\}$

$A = \{1,2,3,4,5,6,....\}$

$A^C = \{-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0\}$

4. Difference: The difference of the sets $A$ and $B$ in this order is the set of elements that belong to $A$ but not to B. Symbolically, the difference of sets are denoted by ‘$-$’
For example, If $A=\{1,2,3,4\}$ and $B=\{4,5,6,8\}$,
Then, $A-B=\{1,2,3\}$ and $B-A=\{5,6,8\}$

Important Formulae of Sets

Properties of Sets

Here’s a concise table summarizing the Properties of Sets with three columns — property name, description, and formula:

List of related topics to sets according to NCERT/JEE MAIN

This section covers all the important topics from sets that are based on NCERT books and frequently asked in JEE Main exams. It will help you focus on the right areas while studying and ensure you don’t miss any essential concepts needed for better preparation. The list is simple, clear, and aligned with the exam pattern.

Solved Examples Based on Set

Example 1: Which of the following are equal sets?
1) $A=\{1,2,3,4\}$ and $B=$ collection of natural numbers less than 6
2) $A=\{$ prime numbers less than 6$\}$ and $B=\{$ prime factors of 30$\}$
3) $A=\{0\}$ and $B=\{x: x>15$ and $x<5\}$
4) $\mathrm{A}=\{5\}$ and $\mathrm{B}=\left\{\mathrm{x}: x^2=25\right\}$

Solution: In option (1), $A=\{1,2,3,4\}$ but $B=\{1,2,3,4,5\}$, hence not equal.
In option (2), $A=\{2,3,5\}$ but $B=\{2,3,5\}$, hence equal.
In option (3), $A=\{0\}$ but $B=$ Null set, hence not equal.
In option (4), $A = {5}$ but $B = {-5, 5}$, hence not equal.
Hence, the answer is the option (2).

Example 2: If a set has $32$ subsets. How many elements does it have?
Solution: As we know, if a set has $n$ elements, it will have $2^n$ subsets.
Thus $2^n=32$

$\Rightarrow n=5$

Hence, the answer is $5$.

Example 3: $\mathbf{A}$ is a set containing $\mathbf{n}$ elements. A subset $P_1$ is chosen and $\mathbf{A}$ is reconstructed by replacing the elements of $P_1$. The same process is repeated for subsets $P_1, P_2, \ldots, P_m$ with $m>1$. The number of ways of choosing $P_1, P_2, \ldots, P_m$, $P_1 \cup P_2 \cup \ldots \cup P_m=A$ so that $P_1 \cup P_2 \cup \ldots \cup P_m=A$ is
1) $\left(2^m-1\right)^{m n}$
2) $\left(2^n-1\right)^m$
3) ${ }^{m+n} C_m$
4) none of these

Solution:
$\operatorname{Let} A=\left\{a_1, a_2, \ldots, a_n\right\}$ for each $a_i(1 \leq i \leq n)$
Either $a_i \in P_j$ or $a_i \notin P_j(1 \leq j \leq m)$
$\therefore$ There are $2^m$ choices in which $a_i, a_j$ belongs to $P_j$,
Also, there is exactly one choice, i.e, $a_i \notin P_j$
$\therefore a_i \in P_1 \cup P_2 \cup \ldots \cup P_m$ in ( $2^m-1$ ) ways.
Since there are n elements in the set A, the number of ways of constructing subsets.
$P_1, P_2, \ldots, P_m$ is $\left(2^m-1\right)^n$

Example 4: If $\mathrm{n}(\mathrm{U})=20, \mathrm{n}(\mathrm{A})=12, \mathrm{n}(\mathrm{B})=9, n(A \cap B)=4$ where U is the universal set, then $n(A \cup B)^c=$

Solution:

$ \begin{aligned} & n(A \cup B)=n(A)+n(B)-n(A \cap B)=12+9-4=17 \\ & n(A \cup B)^{\prime}=n(U)-n(A \cup B)=20-17=3 \end{aligned} $
Hence, the answer is 3.

Example 5: If $A-B=X$ and $A-C=Y$. then the simplification of $A-(B \cup C)$ is
1) $X \cap Y$
2) $X \cup Y$
3) $X-Y$
4) $Y-X$

Solution: $P-Q=P \cap Q^{\prime}$
So, $A - (B \cup C) = A \cap (B \cup C)'$

$A \cap (B' \cap C') = (A \cap B') \cap (A \cap C') = (A - B) \cap (A - C)$

$X \cap Y$
Hence, the answer is option 1.

How to Study Sets?

Start by learning what a set is and how it is written. Be clear about the types of sets in mathematics, and practice operations like union, intersection, and complement. Work on examples to understand how sets are applied in problems.

If you are preparing for competitive exams, practice solving questions rather than memorizing solutions. A strong grip on basic concepts will help you tackle even the trickiest problems easily.

Recommended Books for Sets

This section lists the best books for studying sets, with clear explanations and plenty of practice questions. These books will help you strengthen your concepts and prepare effectively for exams.

Note: Start with NCERT and then choose one of the other books based on your level and comfort. It’s better to follow one reference thoroughly than to switch between multiple books.

NCERT Resources

This section provides NCERT resources for sets, including explanations and practice problems. It’s a great way to build a strong foundation and clear your basics.

NCERT Class 11 Maths Notes for Chapter 1 - Sets

NCERT Class 11 Maths Solutions for Chapter 1 - Sets

NCERT Class 11 Maths Exemplar Solutions for Chapter 1 - Sets

NCERT Subjectwise Resources

This section provides subjectwise NCERT resources to help you study sets topic by topic and practice effectively.

Practice Questions based on Sets

This section includes practice questions to help you apply the concepts of sets. Solving these problems will improve your understanding and prepare you for exams like JEE Main.