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

**SolutionA = { 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

**S**olution

(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