FINDING THE CARDINAL NUMBER OF A SET

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)

Solved Questions

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 and Infinite Set

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 : x2–5x+6 = 0, x ∈N}

Solution :

x2–5x+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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