Page 24 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 1
= n(n + l) + (n + 1)
2
(n 1) (n + 2)
+
=
2
Thus P(n + 1) is also true.
Similarly, by using the Principle of Induction, you can prove that

1
(i) the sum of the squares of the first n natural numbers is n(n + l) (2n +1); and
6
1
2
(ii) the sum of the cubes of the first n natural numbers is n (n + 1) .
2
4
Self Assessment
Choose appropriate answer for the following:
1. The complement of the set S is the set of all those element of ................which do not belong
to S. It is denoted by S.
(a) universal set (b) empty set
(c) union set (d) intersection set

2. Let S and T all two sets. The collection of all elements which belong to S or T is called
.........................
(a) universal (b) union

(c) intersection (d) Difference of two set
3. The intersection .................. of sets S and T is defined to be the set of all those elements
which belong to both S and T.

(a) S  T (b) S  T
(c) S  T (d) S  T
4. If let S = {1, 2, 3} and T = {a, b, c} and let f : S  T be defined as f(1) = a, f(2) = b, f(b) = c. Then
f is.....................
(a) one-one (b) onto
(c) one-one and onto (d) one-one and surjection
5. The set S is called the domain of the function f and T is called its ....................

(a) Range (b) pre-domain
(c) co-domain (d) bijection

1.6 Summary

 We have recalled some of the basic concepts of sets and functions in section 1.2. A set is a
well-defined collection of objects. Each object is an element or a member of the set. Sets are
generally designated by capital letters and the members by small letters enclosed with
braces. There are two ways to indicate the members of a set. The tabular method or listing
method in which we list each element of a set individually and the set-builder method

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