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.