1.4 Finite and Infinite Sets

A set which is empty or consists of a definite number of elements is called finite otherwise, the set is called infinite. By number of elements of a set S, we mean the number of distinct elements of the set and we denote it by \(n(\mathrm{S})\).

(a) A = {1, 2, 3, 4, 5} is a finite set as A contains 5 elements. So we can write \(n(\mathrm{A})\) = 5.
(b) C = { men living presently in different parts of the world} is some finite number.
(c) Let W be the set of the days of the week. Then W is finite.
(d) Let G be the set of points on a line. Then G is infinite.

Note: All infinite sets cannot be described in the roster form. For example, the set of real numbers cannot be described in this form, because the elements of this set do not follow any particular pattern.

Example 6: State which of the following sets are finite or infinite:
(i) \(\{x: x \in \mathrm{N}\) and \((x-1)(x-2)=0\}\)
(ii) \(\left\{x: x \in \mathrm{N}\right.\) and \(\left.x^{2}=16\right\}\)
(iii) \(\{x: x \in \mathrm{N}\) and \(4 x-1=0\}\)
(iv) \(\{x: x \in \mathrm{N}\) and \(x\) is prime \(\}\)
(v) \(\{x: x \in \mathrm{N}\) and \(x\) is odd \(\}\)

Solution: (i) Given set \(=\{1,2\}\). Hence, it is finite.
(ii) Given set \(=\{2\}\). Hence, it is finite.
(iii) Given set \(=\phi\). Hence, it is finite.
(iv) The given set is the set of all prime numbers and since set of prime numbers is infinite. Hence the given set is infinite.
(v) Since there are infinite number of odd numbers, hence, the given set is infinite.