# Cardinal Number of set

In this chapter we will learn the definition of cardinal number of set with method of calculation and solved example.

## What is Cardinal number of set?

The number of distinct element present in a set is called cardinal number of set.

For Example;
Consider the set A with following elements.
A = { 2, 5, 7, 9 }

Note that there are 4 distinct element in the set A.
Hence the cardinal number of set A is 4.

Example 02
Consider the set B with following elements.
B = { 4, 7, 10, 10 }

Here the number of elements present in set B is 4.
But note that the element 10 is repeated twice in the set.

Actually there are 3 distinct element in the above set B.
Hence, the cardinal number of set B is 3.

## Representing cardinal number in Set Theory

The cardinal number is represented by following expressions;

For Example, Consider the below set A;
A = { 3, 5, 7, 9, 11 }

The cardinal number of set A is expressed as;

## Special Case of Cardinal Number

There are two special cases of cardinal numbers which you need to understand.

(i) Cardinal Number of Empty set
(ii) Cardinal Number of Infinite set

### Cardinal Number of Empty Set

Given below is the empty set A.

A = { 𝜙 }

Note that in empty set there is no elements present. Hence the cardinal number of empty set is 0.

Conclusion
The cardinal number of empty set is 0

### Cardinal Number of Infinite set

Infinite set is the set which contains unending number of elements.

Given below is the set A with infinite elements;
A = { x : x is set of positive even numbers }

In Roster Form, the set A can be written as;
A = { 2, 4, 6, 8, 10, 12, . . . . . }

You can see that the set is never ending, so the cardinal number cannot be defined for this set.

Conclusion
The cardinal number of infinite set is not defined.

## Solved Problems on Cardinal Number of Set

(01) Given below are two set A & B.

Find the cardinal number of both set A & B

Solution
A = { 2, 4, 6, 6, 12, 3, 3, 3 }

Note that there are 8 elements in the above set.
But element 6 is repeated twice and number 3 is repeated thrice.

Rewriting the set A with unique elements we get;
A = { 2, 4, 6, 12, 3 }

There are 5 distinct elements in the set A.
Hence, the cardinal number of set A is 5.
n (A) = 5

B = { 6, 1, 9, 3, 6 }

There are 5 elements in set B.
But note that element 6 is repeated twice.

Rewriting the set B with distinct elements.
B = { 6, 1, 9, 3 }

So there are 4 distinct elements in set B.
Hence, the cardinal number of set B is 4.
n ( B ) = 4

(02) Given below are two sets X & Y.

Find the cardinal number of both the sets.

Solution
X = { M, A, T, H, E, M, A, T, I, C, S }

There are 11 elements in the set X.
But there are elements which are repeated more than one.

Letter ” M” is repeated twice.
Latter ” A ” is also repeated twice.
Similarly other letters also repeated in the set X.

Rewriting the set X with distinct elements.

X = { M, A, T, H, E, I, C, S }

Note that there are 8 distinct elements in the above set.
Hence, the cardinal number of set X is 8.
n (X) = 8

Y = { x : x contain letters in word “APPLE” }

Writing the set Y in Roster form
Y = { A, P, P, L, E }

Note that there are 5 elements in the above set, but some of the letters are repeated.

Rewriting the set Y with distinct elements, we get;
Y = { A, P, L, E }

Hence there are 5 distinct elements in set Y.
The cardinal number of set Y is 5.
n ( Y ) = 5

(03) Find the cardinal number of below set A.
A = { x : natural number between 4 & 46 & divisible by 5 }

Solution
Let us first write the set A in Roster form.
A = { 5, 10, 15, 20, 25, 30, 35, 40, 45 }

Note that there are 9 distinct elements in set A.
Hence cardinal number of set A is 9.
n (A) = 9

(04) Find the cardinal number of set X
X = { Africa, North America, Europe, Asia, Antarctica, South America }

Solution
There are 6 distinct element in set X.
Hence the cardinal number of set X is 6.
n (X) = 6

(05) Find the cardinal number of set A & B

Solution
(i) A = { x : x is prime number between 4 & 15 }

Writing the set A in Roster form
A = { 5, 7, 11, 13}

There are 4 distinct element in set A.
Hence cardinal number of set A is 4.
n (A) = 4

(ii) B = { x : x is factor of number 2}

Writing the set B in Roster form.
B = { 1, 2 }

There are two distinct element in set B.
Cardinal number is set B is 2.
n (B) = 2