In this chapter we will learn the difference between finite and infinite sets with properties and examples.

## Finite Set definition

A set is said to be finite set if **its elements can be counted by natural numbers**.

This mean that the number of element of finite set is finite and gets over after certain point.

**Void set** ( which does not contain any element ) is also classified under** finite set**.

### Number of element in finite set

The **number of distinct element** present in given set A is called **order of set A** or** cardinal number** and is denoted by** n (A)**.**For example;**

If there are three distinct element in set A then Order of Set n(A) = 3

### Examples of Finite set with cardinal number

(i) **Y = { x : x is letter in English Alphabet }**

We know that in English Alphabets there are 26 letters.

Hence, the set Y is a finite set as it contains finite number of elements.

The Order of set Y is n(Y) = 26

**(ii) **A = { x : \mathtt{x^{2} =\ 36} }

Solving the expression \mathtt{x^{2} =\ 36} ;

\mathtt{x^{2} =\ 36}\\\ \\ \mathtt{x\ =\ \sqrt{36}}\\\ \\ \mathtt{x\ =\ +6\ ,\ -6}

The set A can also be represented as;

A = { 6, -6 }

Hence, the set A is a finite set as it contain two elements.

Order of set n(A) = 2

(iii) **Y = { x ; x 𝜖 N & x < 100 }**

The above set states that it contains variable x such that;

⟹ x belongs to natural number

⟹ x is less than 100

Hence, all the numbers from 1 to 99 are part of set Y.

The set Y can also be expressed as;

Y = { 1, 2, 3, 4, 5 . . . . . 99 }

So the set Y is a finite set as it contains finite number of elements which ends after certain point.

Also, the order of set Y is 99.**n (Y) = 99**

(iv) **P = { persons living in New Zealand }**

Since there are finite number of people living in New – Zealand and can be easily counted, the above set P is a finite set.

## Infinite Set definition

A set which contains** unending number of elements** are called **infinite set**.

Since the number of infinite set are unending, it is not possible to count the element and give order of the set.

### Examples of Infinite set

(a) **A = { Number of stars in Universe }**

There are infinite numbers of stars in universe. Hence the given set A is infinite set.

(b) ** X = { Set of even natural numbers }**

The set X is an infinite set as it contains unending number of elements.

The set X can also be expressed as;

X = { 2, 4, 6, 8, 10, 12, 14, 16, 18 . . . . }

You can see that the number of elements goes on & on without any end. Hence it’s a perfect example of infinite set.

(c) **Z = { set of all points in a plane }**

In a plane, there can be infinite number of points.

Hence, the set Z is an infinite set.

(d) ** Y = { x ; x 𝜖 N & x > 5 }**

The set states that it contain variable x such that;

⟹ x belongs to natural number N

⟹ x is greater than 5

The above set Y can also be expressed as;

Y = { 6, 7, 8, 9, 10, 11, 12 . . . . }

You can see that set Y contains unending number of elements. Hence the given set is infinite set.