In mathematics, a collection of particular things or a group of particular objects is called a set. The theory of sets was developed by George Cantor and is being used in all branches of mathematics nowadays. It is being used in almost every branch of mathematics from relations and functions to the study of geometry, sequences, probability, etc. According to him ‘A set is a well-defined collection of distinct objects of our perception or of our thought, to be conceived as a whole’.

Let’s understand what is a set in math and the symbols used in its notation along with properties with examples.

## What is a Set in Math?

Set in math is a well-defined collection of objects. A set is represented by a capital letter. The sets are used to represent people, places, objects, letters of the alphabet, numbers, shapes, variables, etc.

Some of the examples of set in math are

- Natural numbers
- Even natural numbers less than $10$, i.e., $2$, $4$, $6$, $8$
- The rivers of India
- The vowels in the English alphabet, viz, $a$, $e$, $i$, $o$, $u$
- Various triangles of triangles
- Factors of $528$, viz., $1$, $2$, $3$, $4$, $6$, $8$, $11$, $12$, $16$, $22$, $24$, $33$, $44$, $48$, $66$, $88$, $132$, $176$, $264$, $528$
- The solution of the equation: $x^2 – 5x + 6 = 0$, viz, $2$ and $3$
- Letters in the word ‘$\text{MATHEMATICS}$’, i.e., $\text{M}$, $\text{A}$, $\text{T}$, $\text{H}$, $\text{E}$, $\text{I}$, $\text{C}$, $\text{S}$

## What is Meant by Well-Defined in Set?

Well-defined means, it must be absolutely clear which object belongs to the set and which does not.

Let’s consider some examples to understand it better.

- ‘The collection of positive numbers less than $10$’ is a set, because, given any number, we can always find out whether that number is a positive number and less than $10$ and hence belongs to the collection or not.
- But ‘the collection of good students in your class’ is not a set as in this case no definite rule is available that you can use to check whether a particular student in your class is good or not and hence is an element of a set or not.
- ‘The collection of students in your class scoring more than $90 \%$ in the latest math test’ is a set, because here also, given any student in your class, you can easily check whether the student has scored more than $90 \%$ in the latest math test or not.

### Examples of Sets

**Example 1:** Which of the following are sets? Justify your answer.

a) The collection of all the months of a year beginning with the letter $\text{J}$.

The collection of all the months of a year beginning with the letter $\text{J}$ is a set since you can clearly check that the months ‘January’, ‘June’, and ‘July’ are the months beginning with the letter $\text{J}$ and hence is well-defined.

b) The collection of the ten most talented writers of India.

The collection of the ten most talented writers of India is not a set, as the word ‘most talented’ does not define properly, which writer from India should be part of a set and which writer should be omitted.

c) A team of eleven best-cricket batsmen of the world.

A team of eleven best-cricket batsmen of the world is not a set, as the word ‘best’ does not define properly, which batsman should be part of a set and which batsman should not be included.

d) The collection of all boys in your class.

The collection of all boys in your class is a set because the word ‘all’ clearly explains which boy in your class should be the part of a set and which boy should be left out.

e) A collection of novels written by the writer Munshi Prem Chand.

A collection of novels written by the writer Munshi Prem Chand is a set as it clearly mentions that novels written by a particular writer – Munshi Prem Chand.

## Terms Associated With Sets

### Element

The items present in a set are called either elements or members of a set. The elements of a set are enclosed in curly brackets separated by commas.

For example, in a set of natural numbers less than $5$, the elements of the set are $1$, $2$, $3$, and $4$.

**Note:** $5$ is not an element of the above set.

### Belongs To

The term ‘belongs to’ is used to indicate that a particular item is an element of the given set. To denote that an element is contained in a set or belongs to the set, the symbol ‘$\in$’ is used.

For example, if the set of natural numbers less than $5$ is represented by $\text{A}$, then

- $1$ belong to $\text{A}$ or $1 \in \text{A}$
- $2$ belong to $\text{A}$ or $2 \in \text{A}$
- $3$ belong to $\text{A}$ or $3 \in \text{A}$
- $4$ belong to $\text{A}$ or $4 \in \text{A}$

### Does Not Belong To

The term ‘does not belong to’ is used to indicate that a particular item is not an element of the given set. To denote that an element is not present in a set or does not belong to the set, the symbol ‘$\notin$’ is used.

For example, if the set of natural numbers less than $5$ is represented by $\text{A}$, then

- $0$ does not belong to $\text{A}$ or $0 \notin \text{A}$
- $-5$ does not belong to $\text{A}$ or $-5 \notin \text{A}$
- ‘$a$’ does not belong to $\text{A}$ or $a \notin \text{A}$
- $0.5$ does not belong to $\text{A}$ or $0.5 \in \text{A}$

### Cardinal Number

The cardinal number, cardinality, or order of a set denotes the total number of elements in the set. For natural even numbers less than $10$, $n(\text{A}) = 4$. Sets are defined as a collection of unique elements. One important condition to define a set is that all the elements of a set should be related to each other and share a common property. For example, if we define a set with the elements as the names of months in a year, then we can say that all the elements of the set are the months of the year.

### Empty or Null Set

A set that does not contain any element is called the empty set or the null set or the void set.

For example, the set of the number of outcomes for getting a number greater than $6$ when rolling a die. As we know, the outcomes of rolling a die are $1$, $2$, $3$, $4$, $5$, and $6$. Thus, in the set with numbers greater than $6$, there will be an empty set. That means there will be no elements and is called the empty set.

## Representation of Sets

There are different set notations used for the representation of sets. They differ in the way in which the elements are listed. The three set notations used for representing sets are:

- Semantic form
- Roster form
- Set Builder form

### Semantic Form

Semantic notation describes a statement to show what are the elements of a set.

For example, Set $\text{A}$ is the list of the first ten natural numbers.

### Roster Form

The most common form used to represent sets is the roster notation in which the elements of the sets are enclosed in curly brackets separated by commas.

For example, Set $\text{A} = {2, 3, 5, 7}$, which is the collection of prime numbers less than $10$. In a roster form, the order of the elements of the set does not matter. The set of prime numbers less than $10$ can also be defined as ${2, 7, 5, 3}$ or ${7, 5, 3, 2}$. Also, if there is an endless list of elements in a set, then they are defined using a series of dots at the end of the last element. For example, infinite sets are represented as, $\text{N} = {1, 2, 3, 4, 5 …}$, where $\text{N}$ is the set of natural numbers.

### Set Builder Form

The set builder notation has a certain rule or a statement that specifically describes the common feature of all the elements of a set. The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.

For example, $\text{A} = { x | x \text{ is an odd number}, x \le 20}$. The statement says, all the elements of set $\text{A}$ are odd numbers that are less than or equal to $20$. Sometimes a “:” is used in place of the “|” ($\text{A} = { x : x \text{ is an odd number}, x \le 20}$).

### Examples of Representation of Sets

**Example 1: **Write the given statement in three methods of representation of a set:

The set of all integers that lies between $-2$ and $7$

The methods of representations of sets are:

- Statement Form: ${ \text{I is the set of integers that lies between } -2 \text{ and }7}$
- Roster Form: $\text{I} = {-1, 0, 1, 2, 3, 4, 5, 6 }$
- Set-builder Form: $\text{I} = { x: x \in \text{I}, -2 \lt x \lt 7 }$

**Example 2:** Find the elements of the sets represented as follows and write the cardinal number of each set.

a) Set $\text{A}$ is the first $8$ multiples of $7$

b) Set $\text{B} = {a,e,i,o,u}$

c) Set $\text{C} = {x | x \text{ are even numbers between } 20 \text{and | 40}$

a) Set $\text{A} = {7,14,21,28,35,42,49,56}$. These are the first $8$ multiples of $7$.

Since there are $8$ elements in the set, cardinal number $n(\text{A}) = 8$.

b) Set $\text{B} = {a,e,i,o,u}$. There are five elements in the set,

Therefore, the cardinal number of set $\text{B}$, $n(\text{B})$ = 5$.

c) Set $\text{C} = {22,24,26,28,30,32,34,36,38}$. These are the even numbers between $20$ and $40$, which make up the elements of set $\text{C}$.

Therefore, the cardinal number of set $\text{C}$, $n(\text{C}) = 9$.

**Example 3:** Express the given set in set-builder form: $\text{A} = {2, 4, 6, 8, 10, 12, 14}$

The given set is $\text{A} = {2, 4, 6, 8, 10, 12, 14}$

Using sets notations, we can represent the given set $\text{A}$ in set-builder form as,

$\text{A} = {x | x \text{ is an even natural number less than }15}$

**Example 4:** Write the solution set of the equation $x^2 – 4=0$ in roster form.

$x^2 – 4 = x^2 – 2^2 = (x – 2) (x + 2)$

Therefore, $x = 2, -2$

Thus, $\text{A} = {-2, 2}$.

**Example 5:** Write the set $\text{A} = {1, 4, 9, 16, 25, . . . }$ in set-builder form.

If we see the pattern here, the numbers are squares of natural numbers, such as:

$1^2 = 1$

$2^2 = 4$

$3^2 = 9$

$4^2 = 16$

And so on.

$\text{A} = {x : x \text{ is the square of a natural number }}$

Or we can write;

$\text{A} = {x : x = n^2 , \text{where } n \in \text{N}}$

**Example 6:** Write the interval $(6, 12)$ in set builder form.

Let $\text{A}$ be the interval $(6, 12)$.

The interval $(6, 12)$ in set builder form is $\text{A} = {x: x \in \text{R}, 6 \lt x \lt 12}$

## Properties of Sets

Similar to numbers, sets also have properties like associative property, commutative property, and so on. There are six important properties of sets. Given, three sets $\text{A}$, $\text{B}$, and $\text{C}$, the properties for these sets are as follows.

**Commutative Property:**It states that the union of sets $\text{A}$ and $\text{B}$ is equal to the union of sets $\text{B}$ and $\text{A}$, i.e., $\text{A} \cup \text{B} = \text{B} \cup \text{A}$**Associative Property:**It states that the order of sets involved in the union does not alter the contents of the resultant set, i.e., $(\text{A} \cap \text{B}) \cap \text{C} = \text{A} \cap (\text{B} \cap \text{C})$.**Distributive Property:**It states that the union of sets is distributive over the intersection of sets, i.e., $\text{A} \cup (\text{B} \cap \text{C}) = (\text{A} \cup \text{B}) \cap (\text{A} \cup \text{C})$.**Identity Property:**It states that the union of a set with a null set results in the same set, i.e., $\text{A} \cup \phi = \text{A}$.**Complement Property:**It states that the union of a set and its complement is a universal set, i.e., $\text{A} \cup \text{A}’ = \text{U}$.**Idempotent Property:**It states that the union or intersection of a set with itself gives the same set, i.e., $\text{A} \cap \text{A} = \text{A}$ and $\text{A} \cup \text{A} = \text{A}$.

## Practice Problems

- The symbol used to denote an element of a set is ___
- $\in$
- $\notin$
- $\cup$
- $\cap$

- Write the set $\text{A} = {1, 2, 3, 4, 5, …}$ in set-builder form.
- Write the set-builder form of set $\text{X} = {3,9,27,81}$.
- Represent the set $\text{Y} = {x : x \text{ is an integer}, –1 \lt x \lt 5}$ in roster form.
- What is the set builder form of: $(-3, 0)$ and $(6,12)$.
- A set $\text{S} = {5, 10, 15, 20, 25, 30}$ is given in the roster form. Rewrite it in,
- set builder notation, and
- statement method

## FAQs

### What is set? Give an example.

Set in math is a well-defined collection of objects. A set is represented by a capital letter. The sets are used to represent people, places, objects, letters of the alphabet, numbers, shapes, variables, etc.

Examples of sets are

a) Even natural numbers less than $10$, i.e., $2$, $4$, $6$, $8$

b) The rivers of India

c) The vowels in the English alphabet, viz, $a$, $e$, $i$, $o$, $u$

d) Various triangles of triangles

e) Letters in the word ‘$\text{MATHEMATICS}$’, i.e., $\text{M}$, $\text{A}$, $\text{T}$, $\text{H}$, $\text{E}$, $\text{I}$, $\text{C}$, $\text{S}$

### Why do we use sets in Maths?

The purpose of using sets is to represent the collection of relevant objects in a group. In maths, we usually represent a group of numbers like a group of natural numbers, a collection of real numbers, etc.

### What are the elements of sets?

The items present in a set are called either elements or members of a set. The elements of a set are enclosed in curly brackets separated by commas.

For example, in a set of factors of $12$, the elements of the set are $1$, $2$, $3$, $4$, $6$, and $12$.

### What is set in simple word?

A set is a group of things that belong together, like the set of even numbers $(2,4,6…)$ or the bed, nightstands, and dresser that make up your bedroom set. Set has many different meanings. As a verb, it means to put in place.

### How are sets named?

Sets are named and represented using capital letters such as $\text{A}$, $\text{B}$, $\text{C}$, etc. In the set theory, the elements that a set comprises can be any kind of thing: people, letters of the alphabet, numbers, shapes, variables, etc.

## Conclusion

A set in math is a well-defined collection of objects. A set is represented by a capital letter. The sets are used to represent people, places, objects, letters of the alphabet, numbers, shapes, variables, etc. The three set notations used for representing sets are semantic form, roster form, and set builder form.