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 x\(^{2}\) < 30}
(ii) Q = {x | x is a factor of 20}
Solution:
(i) Given, P = {x | x ∈ N and x\(^{2}\) < 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
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