FINITE AND INFINITE SETS

Finite Set

If the number of elements in a set is zero or finite, then the set is called a finite set.

For example,

(i)  Consider the set A of natural numbers between 8 and 9.

There is no natural number between 8 and 9.

So, A  =  {  } and n(A) = 0.

Hence, A is a finite set.  

(ii)  Consider the set X = {x : x is an integer and -1 ≤ x ≤ 2}

So, X  =  {-1, 0, 1, 2} and n(X)  = 4  

Hence, X is a finite set. 

Note : 

The cardinal number of a finite set is finite.

Infinite Set

A set is said to be an infinite set, if the number of elements in the set is not finite.

For example, 

Let W  =  The set of all whole numbers .

That is, W  =  {0, 1, 2, 3,......................}

The set of all whole numbers contain infinite number of elements.

Hence, W is an infinite set. 

Note : 

The cardinal number of an infinite set is not a finite number. 

Solved Problems

Identify the following sets as finite or infinite.

Problem 1 :

Set of vowels in English alphabet finite set

Solution :

Finite set

Roster form of the given set :

{a, e, i, o, u}

In the above set, there are five elements.

Because the above set has a finite number of elements, it is a finite set.  

Problem 2 :

Set of even numbers from 1 to 10

Solution :

Finite set

Roster form of the given set :

{2, 4, 6, 8, 10}

In the above set, there are five elements.

Because the above set has a finite number of elements, it is a finite set.  

Problem 3 :

Set of natural numbers

Solution :

Infinite set

Roster form of the given set :

{1, 2, 3, 4, 5,................}

In the above set, the number of elements is infinite. 

Because there are infinite natural numbers, the set of natural numbers is an infinite set. 

Problem 4 :

Set of integers

Solution :

Infinite set

Roster form of the given set :

{................-3, -2, -1, 0, 1, 2, 3, 4, 5,................}

In the above set, the number of elements is infinite. 

Because there are infinite number of integers, the set of integers is an infinite set.

Problem 5 :

A = {4, 5, 6, .............}

Solution :

We cannot count the number of elements in the given set. 

Hence it is infinite set.

Problem 6 :

B = {0, 1, 2, 3, 4, ......................,75}

Solution :

The given set has countable number of elements.

Hence it is finite set.

Problem 7 :

X = {x : x is an even natural number}

Solution :

Since the given set has even natural number, it starts with 2.

X = {2, 4, 6, ...............}

We couldn't find the number of elements in the given set.

Hence it is infinite set.

Problem 8 :

Y = {x : x is a multiple of 6 and x > 0}

Solution :

The elements of the given set set should be multiples of 6.

Y = {6, 12, 18, 24, .....................}

We couldn't find the number of elements in the given set. Hence it is infinite set.

Problem 9 :

P = The set of letters in the word ‘freedom’

Solution :

The set P contains letters of the word "freedom"

P = { f, r, e, d, o, m }

Set P has countable number of elements.

Hence it is finite set.

Problem 10 :

The set of roots of the equation x² - 3x + 2 = 0

Solution :

Let A be the set contain the roots of the equation. 

Solving the equation x² - 3x + 2 = 0, we get

x = 1, 2

So, A = {1, 2}.

Since the set A contains only two elements, it is finite set.

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