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