Finite and Infinite Sets (Set theory part 3 NCERT)

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Finite and Infinite Sets

Let   A = {1, 2, 3, 4, 5},   B = {a, b, c, d, e, g}
and   C = {men living presently in different parts of the world}

We observe that A contains 5 elements and B contains 6 elements. How many elements does C contain? As it is, we do not know the number of elements in C, but it is some natural number which may be quite a big number. By number of elements of a set S, we mean the number of distinct elements of the set and we denote it by n (S). If n (S) is a natural number, then S is non-empty finite set

Consider the set of natural numbers. We see that the number of elements of this set is not finite since there are infinite number of natural numbers. We say that the set of natural numbers is an infinite set. The sets A, B and C given above are finite sets and n(A) = 5, n(B) = 6 and n(C) = some finite number.

Definition 2 A set which is empty or consists of a definite number of elements is called finite otherwise, the set is called infinite
Consider some examples :

(i) Let W be the set of the days of the week. Then W is finite.
(ii) Let S be the set of solutions of the equation x2–16 = 0. Then S is finite.
(iii) Let G be the set of points on a line. Then G is infinite.

When we represent a set in the roster form, we write all the elements of the set within braces { }. It is not possible to write all the elements of an infinite set within braces { } because the numbers of elements of such a set is not finite. So, we represent some infinite set in the roster form by writing a few elements which clearly indicate the structure of the set followed ( or preceded ) by three dots.

For example, {1, 2, 3 . . .} is the set of natural numbers, {1, 3, 5, 7, . . .} is the set of odd natural numbers, {. . .,–3, –2, –1, 0,1, 2 ,3, . . .} is the set of integers. All these sets are 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 ∈N and (x – 1) (x –2) = 0}
(ii) {x : x ∈N and x2 = 4}
(iii) {x : x ∈N and 2x –1 = 0}
(iv) {x : x ∈N and x is prime}
(v) {x : x ∈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 = φ. 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.