When a set is finite, it is very useful to know how many elements it has. The number of elements in a set is called the Cardinal number of the set.
The cardinal number of a set A is denoted by n(A)
Question 1 :
Find the cardinal number of the following sets.
(i) M = {p, q, r, s, t, u}
Solution :
Number of elements in the set is 6. Hence n(M) = 6.
(ii) P = {x : x = 3n + 2, n∈W and x < 15}
Solution :
Since n belongs to whole number, we have to start with 0.
By applying the values of n from 0 to 14, we get 15 different values for x. Hence n(P) is 15.
(iii) Q = { y : y = 4/3n, n ∈ N and 2 < n ≤ 5}
Solution :
The values of n are 3, 4, 5. By applying the above three values for n, we get different values of y. Hence n(Q) is 3.
(iv) R = {x : x is an integers, x ∈ Z and –5 ≤ x < 5}
Solution :
The elements of R are
R = {-5,-4, -3, -2, -1, 0, 1, 2, 3, 4}
n(R) = 10
(v) S = The set of all leap years between 1882 and 1906.
Solution :
The leap years 1884, 1888, 1892, 1896, 1900, 1904.
Hence n(S) = 6
Finite set :
A set with finite number of elements is called a finite set.
Infinite set :
A set which has infinite number of elements is called an infinite set.
Question 2 :
Identify the following sets as finite or infinite.
(i) X = The set of all districts in Tamilnadu.
Solution :
Districts in Tamilnadu is countable. Hence it is finite set.
(ii) Y = The set of all straight lines passing through a point.
Solution :
We may draw an infinite number of lines through a point.
Hence it is infinite set.
(iii) A = { x : x ∈ Z and x < 5}
Solution :
Z means integers. The elements of A are 1, 2, 3, 4.
Hence set A is finite.
(iv) B = {x : x^{2}–5x+6 = 0, x ∈N}
Solution :
x^{2}–5x+6 = 0
(x - 2) (x - 3) = 0
x = 2 and x = 3
By solving the quadratic equation, we get two different values. Hence B is finite set.
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