# Types of Sets || Empty Set, Finite & Infinite Set, Singleton Set

In this chapter we will learn different types of sets with properties & examples.

To understand the concept you should have basic knowledge of sets and its representation.

## Different Types of Sets

Given below are some of the types of sets with examples.

### Null Set

A set which do not contain any element is known as null set.

⟹ Null set is also called Empty set or Void set.

⟹ It is represented by symbol “𝜙” (known as “Phi”)

⟹ In roster form null set is represented as { }
Note that there is no element inside the curly bracket.

#### Examples of Empty Set

Example 01
A = { x : 2< x < 3 & x 𝜖 N }

Explanation:
Given above is the set with name A.

The set contains element x such that;
⟹ x lies between 2 and 3, and;
⟹ x is a natural number

Since there is no natural number between 2 & 3, the above set A is an empty set.

i.e. A = { }

Example 02
B = { x : x > 2 & x is even prime number }

Explanation
Given above is the set with name B.

Set B contain element x such that;
⟹ x is greater than 2
⟹ x is even prime number

There is no even prime number greater than 2.
Hence set B is a null set.

Example 03
Check if the below set is null set or not?
A = { 0 }

Explanation
The above set contains number 0 in it.
Since there is one element present, it is not a null set.

### Singleton Set

A set containing one element is known as Singleton set.

⟹ The set contains exactly one element, not less not more.

#### Examples of Singleton Set

Example 01
Y = { 9 }

Explanation
Given above is a set named Y containing one element 9.

Since the set contains one element, it is a singleton set.

Example 02
Y = { x : x is a even prime number}

Explanation
Given above is the set named Y.

Set Y contains element x such that;
⟹ x is a even prime number

Since 2 is the only even prime number in the number system, the set can be written as:
Y = { 2 }

Hence, set Y is a singleton set.

Example 03
M = { x : x < 2 & x 𝜖 N }

Explanation
Given above is the set with name M.

Set M contain element x such that;
⟹ x is less than 2
⟹ x belongs to natural number

Since 1 is the only natural number which is less than 2, the set can be written as:
M = { 1 }

There is only one element in the set M.
Hence it is a singleton set.

### Finite Set

A set whose elements can be easily counted is called Finite set

⟹ the listing of elements in finite set is limited and terminates at certain countable number.

⟹ the empty set is also a type of finite set.

⟹ the elements of finite set can be easily expressed in roster form

#### Examples of Finite set

Example 01
A = { 5, 7, 9, 10, 15, 26, 35 }

Explanation
The above set A is a finite set since it contain limited elements.
There are total of 7 elements in set A.

Example 02
A = { Set of Letters in English Alphabets }

Explanation
Set A is a finite set as we know there are 26 letter in English Alphabets.
Since the number of alphabet is ending and easily countable, the set is finite.

Example 03
A = { Number of people in Football stadium }

Explanation
Set A is a finite set since the people are limited and can be counted easily.

Example 04
M = { x : x < 5 & x 𝜖 W }

Solution
Given above is the set named M.

The set M contains element x such that;
⟹ x is less than 5
⟹ x belongs to whole numbers

The set can be written as:
M = { 0, 1, 2, 3, 4 }

Hence, set M is a finite set.

### Infinite Set

A set which contains uncountable and never ending elements is called infinite set.

⟹ Infinite sets contain unending elements

⟹ In infinite sets, the elements cannot be listed by natural numbers.
Since, there are infinite elements, the listing will go on and on and cannot be counted.

⟹ Infinite sets are difficult to represent in roster form.
Generally set builder form of representation is used for infinite sets.

#### Examples of Infinite set

Example 01
A = { Set of positive even numbers}

Explanation
We know that there are infinite number of positive even numbers.

Some examples of +ve even numbers are:
2, 4, 6, 8, 10, 12 . . . . . . . .

The numbers will go on and on.

Since the set contains unending elements, it is an infinite set.

Example 02
X = { Number of stars in sky }

Explanation
There are uncountable stars in the sky.
Hence, the set X is infinite set.

Example 03
Y = { x : x > 15 & x 𝜖 N }

Explanation
Given above is the set named Y.

Set Y contains element x such that;
⟹ x is greater than 5, and;
⟹ x belongs to natural number N

So the possible elements in set Y are;
Y = { 16, 17, 18, 19, . . . . .}

There are uncountable elements in the set Y.
Hence, set Y is an infinite set.

### Equivalent Sets

Two sets are said to be equivalent if their cardinal numbers are same.

Cardinal Numbers is the number of unique element present in the given set.

⟹ Equivalent sets have same number of unique elements.

⟹ The elements can be different. The total number of unique element must be same.

⟹ If sets A & B are equivalent sets then; n (A) = n(B)
i.e. number of element in A = number of element in B.

#### Examples of Equivalent Sets

Example 01
A = { 3, 4, 7 }
B = { 9, 15, 33 }

Both sets A & B are equivalent sets.

Explanation
Set A contains three elements.
n (A) = 3

Set B also contains 3 elements.
n (B) = 3

We get, n(A) = n(B)
Hence, the sets A & B are equivalent sets.

Example 02
Y = { x : 3<x<10 & x 𝜖 N }
Z = { a : 12<a<19 & a 𝜖 N }

Both sets Y & Z are equivalent sets.

Explanation
Y = { x : 3<x<10 & x 𝜖 N }

Given above is set Y which contains element x such that;
⟹ x lies between 3 and 10
⟹ x belongs to natural number N

So the elements of set Y are:
Y = { 4, 5, 6, 7, 8, 9}

There are total of 6 elements.
n(Y) = 6

Z = { a : 12<a<19 & a 𝜖 N }

This is another set Z which contains element a such that;
⟹ a lies between 12 and 19
⟹ a belongs to natural number N

Elements of set Z are;
Z = {13, 14, 15, 16, 17, 18}

There are 6 elements in set Z.
n(Z) = 6

Since, n(Y) = n(Z) = 6, both the sets are equivalent sets.

### Equal Sets

Two sets are said to be equal if both the sets contain same elements.

⟹ If A & B are equal sets then;
All elements of set A is present in set B & vice-versa.

⟹ if set A & B are equal then we write, A = B

#### Example of Equal Sets

Example 01
A = { 6, 10, 13, 7 }
B = { 6, 7, 10, 13 }

The above sets A & B are equal sets.

Explanation:
Set A is made of element 6, 7, 10, 13.

Set B is also made of same elements 6, 7 , 10 , 13.

Hence, both the sets are equal (i.e. A = B )

Example 02
Set A = { 21, 19, 21, 19 }
Set B = { 19, 21 }

Both sets A & B are equal.

Explanation

Set A is made of elements 19 & 21.

Set B is also made of elements 19 & 21.

Since set A & B are made of same elements, they are said to be equal.