The number of elements in a set is called the cardinal number of the set.
Let A be a set containing finite number of elements.
Then the cardinal number of the set A is denoted by n(A).
Find the cardinal number of the following sets.
Example 1 :
A = {-1, 0, 1, 2, 3, 4, 5, 6}
Solution :
Number of elements in the given set is 7.
So, cardinal number of set A is 7.
That is n (A) = 7.
Example 2 :
A = {x : x is a prime factor of 12}
Solution :
To find the cardinal number of the given set, we have to count the number of elements of the set.
So, first we have to list out the elements of the set.
Factor of 12 is 1, 2, 3, 4, 6 and 12.
But prime factor of 12 are 2 and 3.
A = {2, 3}
The given set has 2 elements. Hence n(A) = 2.
Example 3 :
B = {x : x ∈ W, x ≤ 5}
Solution :
Here W stands for whole numbers.
The set contains whole numbers which are less than or equal to 5.
B = {0, 1, ,2, 3, 4, 5}
Set B has 6 elements. Hence n(B) = 6.
Example 4 :
C = {x : x = 5^{n}, n ∈ N and n < 5}
Solution :
We may find the elements of the above set using the pattern x = 5^{n}.
n = 1 x = 5^{n } x = 5^{1} x = 5 |
n = 2 x = 5^{n } x = 5^{2} x = 25 |
n = 3 x = 5^{n} x = 5^{3} x = 125 |
n = 4 x = 5^{n} x = 5^{4} x = 625 |
Example 5 :
D = {x : x is a consonants in English alphabets}
Solution :
In English alphabets other than vowels is known as consonants.
D = { b, c, d, f, g, h, j, k , l, m, n, p, q, r, s, t, v, w, x, y, z}
Total number of elements in set D is 21.
Hence, n (D) = 21
Example 6 :
E = {x : x is even prime number}
Solution :
A number which is divisible by 1 and itself is known as prime number. Among prime numbers there is only one even prime number
That is,
E = { 2 }
Total number of elements in set E is 1.
Hence, n (E) = 1
Example 7 :
E = {x : x < 0, x ∈ W}
Solution :
A number which is less than zero is negative number and it will not be a whole number.
So, the above set will not contain any elements.
Hence n (E) = 0.
Example 8 :
Q = { x : - 3 ≤ x ≤ 5, x ∈ Z }
Solution :
The elements of the set Q lies between -3 and 5.We may take -3 and 5 also.
Q = { -3, -2, -1, 0, 1, 2, 3 }
Number of elements in the above set is 7.
Hence n (Q) = 7.
Finite SetIf the cardinal number of a set is zero or finite, then the set is called a finite set. |
Infinite SetIf a set contains infinite number of elements, then it is said to be an infinite set. |
(i) Consider the set A of natural numbers between 8 and 9.
There is no natural number between 8 and 9.
So, A = { } and n(A) = 0.
Hence, A is a finite set.
(ii) Consider the set X = {x : x is an integer and -1 ≤ x ≤ 2}
So, X = {-1, 0, 1, 2} and n(X) = 4
Hence, X is a finite set.
(i) Consider the set of whole numbers
That is, W = {0, 1, 2, 3,.............}
The set of all whole numbers contain infinite number of elements.
Hence, W is an infinite set.
(ii) Consider the set of natural numbers.
That is, N = {1, 2, 3,.............}
The set of all natural numbers contain infinite number of elements.
Hence, N is an infinite set.
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