Cardinal Number of a Set

What is
the cardinal number of a set?

The number of distinct elements in a finite set is
called its cardinal number. It is denoted as n(A) and read as ‘the number of
elements of the set’.

For example:

(i) Set A = {2, 4, 5, 9, 15} has 5 elements.

Therefore, the cardinal number of set A = 5. So, it is denoted as n(A) = 5.

(ii) Set B = {w, x, y, z} has 4 elements.

Therefore, the cardinal number of set B = 4. So, it is denoted as n(B) = 4.

(iii) Set C = {Florida, New York, California} has 3 elements.

Therefore, the cardinal number of set C = 3. So, it is denoted as n(C) = 3.

(iv) Set D = {3, 3, 5, 6, 7, 7, 9} has 5 element.

Therefore, the cardinal number of set D = 5. So, it is
denoted as n(D) = 5.

(v) Set E = {   }
has no element.

Therefore, the cardinal number of set D = 0. So, it is
denoted as n(D) = 0.

Note:

(i) Cardinal number of an infinite set is not defined.

(ii) Cardinal number of empty set is 0 because it has no
element.

Solved
examples on Cardinal number of a set:

1. Write the cardinal
number of each of the following sets:

(i) X = {letters in the word MALAYALAM}

(ii) Y = {5, 6, 6, 7, 11, 6, 13, 11, 8}

(iii) Z = {natural numbers between 20 and 50, which are
divisible by 7}

Solution:

(i) Given, X = {letters in the word MALAYALAM}

Then, X = {M, A, L, Y}

Therefore, cardinal number of set X = 4, i.e., n(X) = 4

(ii) Given, Y = {5, 6, 6, 7, 11, 6, 13, 11, 8}

Then, Y = {5, 6, 7, 11, 13, 8}

Therefore, cardinal number of set Y = 6, i.e., n(Y) = 6

(iii) Given, Z = {natural numbers between 20 and 50, which
are divisible by 7}

Then, Z = {21, 28, 35, 42, 49}

Therefore, cardinal number of set Z = 5, i.e., n(Z) = 5

2. Find the cardinal
number of a set from each of the following:

(i) P = {x | x ∈ N and x2

2 < 30}

(ii) Q = {x | x is a factor of 20}

Solution:

(i) Given, P = {x | x ∈ N and x2

2 < 30}

Then, P = {1, 2, 3, 4, 5}

Therefore, cardinal number of set P = 5, i.e., n(P) = 5

(ii) Given, Q = {x | x is a factor of 20}

Then, Q = {1, 2, 4, 5, 10, 20}

Therefore, cardinal number of set Q = 6, i.e., n(Q) = 6