1.4 Finite and Infinite Sets

Finite Set

A set having finite number of elements is called a finite set. By the number of elements of a set S, we mean the number of distinct elements of the set, and we denote it by \(n(\mathrm{S})\).
For example,
(i) \(A=\{1,2,3\}\)
Here, \(A\) is a finite set as it has 3 (finite number) elements.
(ii) \(B=\) Set of vowels of English Alphabet \(=\{\mathrm{a}, \mathrm{c}, \mathrm{i}, \mathrm{o}, \mathrm{u}\}\)
Here, \(B\) is a finite set with 5 elements.
(iii) \(C=\) Set of all rivers in India.
However, the list is long, but the number is finite. So, \(C\) is a finite set.
(iv) A = {1, 2, 3, 4, 5} is a finite set as A contains 5 elements. So we can write \(n(\mathrm{A})\) = 5.
(v) C = { men living presently in different parts of the world} is some finite number.
(vi) Let W be the set of the days of the week. Then W is finite.

Infinite Set

A set is said to be an infinite set if the number of elements in the set is not finite.

For example,
(i) \(N=\) Set of all positive integers \(=\{1,2,3,4, \ldots\}\)
There are infinite positive integers and hence, \(N\) is an infinite set.
Similarly, sets \(W, Z, Q\) and \(R\) are all infinite sets.
(ii) \(A=\) Set of all points on a straight line
Since there are infinite points on a straight line, set \(A\) is an infinite set.

Example 7: 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.