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

Set theory is a branch of mathematics dealing with the characteristics of well-defined collections of objects that may or may not be mathematical in nature, such as numbers or functions. Sets are very fundamental concepts which have applications across various domains like statistics, calculus, computer science, etc. This article is about the concept of sets in mathematics. Sets chapter 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.

In our daily life, we often deal with collections of objects like the collection of books, coins, fruits, stationeries, etc. Set is a mathematical way of representing the collections of objects. These collections are made based on specific characteristics. For instance, let us consider the set of books, here being a book is the characteristic. Now, let us form a set consisting of mathematics books. In this case, the mathematics book is the characteristic. This characteristic of the object can be said as the definition of the object.

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

There are two ways of representing a set,

• Roster or tabular form

• 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'.

Types of Set

The types of sets in mathematics are,

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 $n(A) = n(B)$.

Eg. 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 are said to be disjoint if there is no common element between them. (i.e) Two sets are said to be disjoint if each set has distinct elements in it.

Eg. 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.

Eg. 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.

Eg. 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 that can be performed on sets are,

1. Union: Union of two sets combines the values in both the 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 $B-A=\{5,6,8\}$

Properties of Sets

1. A set is well-defined if it is possible to determine if the object belongs to the set or not.

2. It is unordered If it's not in the order. For example, the set $\{1,2,3\}$ is the same as $\{3,2,1\}$.
3. A set cannot have duplicate elements.

List of Topics According to NCERT/JEE MAIN

Solved Examples Based on Set

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

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 colours are not well-defined as it is different for different people. But, the list of all colours in the rainbow is definite and well-defined. So, it is a set.

We can decide with respect to any colour, 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 colours.

Hence, the answer is option 3.

Example 2: 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) $C=\{x: x$ 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 $2=\{2,3,5,7, \ldots .$.$\}$ : 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 option 2.

Example 3: Which of the following sets is different 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 option 3.

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 a class above the age of $15$.
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 option 3.

Importance of Sets class 11

The concept of a set is a fundamental aspect of modern mathematics. Today, practically every discipline of mathematics employs this concept. The ideas of relations and functions are defined using sets. The study of geometry, sequences, probability, and other subjects necessitates the understanding of sets. Georg Cantor, a German mathematician, invented the theory of sets (1845-1918). He originally came upon sets while working on "trigonometric series problems." We'll go over some basic set definitions and operations in this chapter.

Although, in the JEE test, there is just one question from Sets.

However, it is still important.

Set theory is as a topic is not very important but when its use comes in functions and relations then it becomes a very important and basic concept.

NCERT Notes Subject Wise Link:

• NCERT notes Class 11 Maths

• NCERT notes Class 11 Physics

• NCERT notes Class 11 Chemistry

• NCERT notes Class 11 Biology

NCERT Solutions Subject wise link:

• NCERT solutions for class 11 Physics.

• NCERT solutions for class 11 Chemistry.

• NCERT solutions for class 11 Mathematics.

• NCERT solutions for class 11 Biology.