Sets in math can be defined as a collection or group of particular objects which are distinct to each other.

The members of the set are one of a kind, they are unique in nature. The objects present in a set are called the elements of the set and they are included only once in a set.

For example types of fingers in our hand $\{ index,\ middle,\ ring,\ pinky \}$

## Definition

Definition of sets as quoted by the founder of sets Georg Cantor is as follows

“A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thought—which are called elements of the set.”

The different types of sets on the basis of the properties of sets are as follows:

a) Empty set or null set
b) Singleton set
c) Finite set
d) Infinite set
e) Cardinal number of a set
f) Equivalent set
g) Equal set

## Empty Set or Null Set

A set which has no elements in it is called an empty set. It does not contain any objects in it and is thus also known as the null set or void set. The symbol used to represent an empty set is $\phi$ and it is pronounced as phi. Another way to represent the null set is in the roster form $\{\ \}$. An empty set is a finite set as there are $0$ elements in it which is finite in nature. Cardinal number of a null set is $0$

Examples of empty set are as follows:

a) Set of natural numbers less than $0$. We know that natural numbers are those which start with $1$ and keeps on increasing $2,\ 3,\ 4.....$ Infinity. But numbers less than $0$ shall give us all negative numbers which violates the definition of natural numbers. Hence, it is an empty set or null set or void set

b) Dogs with six legs. It is known to us that dogs possess four legs only so the given example of six legs does not make any sense and hence is a null or void set

## Singleton Set

Set which contains only one element in it is called a Singleton set.

Examples of singleton set are as follows:

a) If $P$ = $\{\ x\ |\ x$ is a prime number between $25$ and $30\}$ then $P$ = $\{ 29 \}$. There is a single prime number between $25$ and $30$ which is $29$, so the set $P$ is said to be a singleton set

b) Suppose $A$ = $\{ x\ :\ x$ is a even prime number$\}$ then $A$ is a singleton set as there is only one number which is both even and prime and that is $2$

c) $R$ = $\{ x\ :\ x\ \epsilon\ N$ and $x^{2}$ = $9 \}$
There are two numbers $3$ and -$3$ whose square gives us $9$ but as per the condition mentioned in the set it has to be a natural number. So, there is a single number $3$ whose square is $9$. Thus, the set $R$ is a singleton set.

## Finite Set

The set which contains limited number of elements that can be counted is called a finite set. Empty set is also considered to be a finite set

Examples of finite set are as follows:

a) Set of vowels in English alphabet.
There are five vowels $A,\ E,\ I,\ O,\ U$ out of $26$ letters in English alphabet. As it is a definite set of five elements which can be counted so the set is said to be a finite set

b) $Q$ = $\{ x\ :\ x\ \epsilon\ W,\ x\ <\ 8 \}$
The set $Q$ contains whole numbers less than $8$. They are $0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7$. There are limited $8$ such elements in the set $Q$, hence set $Q$ is said to be a finite set

c) Number of sides in a coin
There are only two sides head and tail appearing on the both sides of the coin. As it is a definite set of two elements, hence the set is said to be a finite set

## Infinite Set

The set which contains unlimited number of elements which is countless and never ending is said to be an infinite set.

Examples of an infinite set are as follows:

a) Number of stars in the sky
There are innumerable numbers of stars present in the sky which cannot be counted, hence the set is limitless and thus said to be an infinite set

b) $A$ = $\{ x\ :\ x\ \epsilon\ N,\ x\ >\ 5 \}$
The set of natural numbers greater than $5$ are $6,\ 7,\ 8,\ 9,\ 10,$..... till infinity. As there is infinite number of natural numbers greater than $5$, hence the set is said to be an infinite set

c) Number of cells making up the human skin. This is also an example of infinite set as innumerable number of living cells make up the human skin, thus infinite set

## Cardinal Number of a Set

Cardinal number of a set is the number which is used to count the elements present in a finite set. It is represented by the letter $n$ with a parenthesis of the name of the set

Examples of cardinal number of a set:

a) Cardinal number of colors in a rainbow. The set representing the seven colors present in a rainbow are $A$ = $\{ violet,\ indigo,\ blue,\ green,\ yellow,\ orange,\ red \}$. So, the cardinal number of rainbow colors is $n(A)$ = $7$

b) Cardinal number of multiples of $5$ till $50$. The set representing the multiples of $5$ till $50$ is $A$ = $\{ 5,\ 10,\ 15,\ 20,\ 25,\ 30,\ 35,\ 40,\ 45,\ 50 \}$. So, the cardinal number of multiples of $5$ would be $n(A)$ = $10$

## Equivalent Set

Equivalent set is defined as a condition when the cardinal number of one set is the same as the cardinal number of another set in comparison. Then, the two sets are said to be equivalent and is represented by the double headed arrow symbol “$\leftrightarrow$”

Examples of equivalent set are as follows:

a) Cardinal number of sides in a pentagon and cardinal number of vowels present in English alphabet. Both sets $A$ and $B$ have five elements. So, as $n(A)$ = $n(B)$ they are said to be equivalent set represented by $A \leftrightarrow B$

b) Cardinal number of angles in a square and cardinal number of primary operators in math. There are four corners each having $90$ degree so $n(A)$ = $4$. Similarly, primary operators in math are $+,\ -,\ *,\ /$ having cardinal number $n(B)$ = $4$. So, as $n(A)$ = $n(B)$ they are said to be equivalent set represented by $A \leftrightarrow B$

## Equal Set

Two sets $P$ and $Q$ are said to be equal sets when same elements of set $P$ are present in set $Q$ and same elements of set $Q$ are present in set $P$. The elements present in both the sets need to be identical.

Examples of equal set are as follows:

a) Set $A$ = $\{ a,\ b,\ c,\ d \}$ and set $B$ = $\{ b,\ d,\ c,\ a \}$. As, we see both sets A and B contains the same elements so, they are said to be equal set

b) Set $P$ = $\{ red,\ blue,\ green,\ yellow \}$ and set $Q$ = $\{ green,\ red,\ blue,\ yellow \}$. As, we see both sets $A$ and $B$ contains the same elements so, they are said to be equal sets.