Page 48 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 3. Show that the set {0, 1} forms a field under the operations ‘+’ and ‘.’ defined by the following
tables:

+ 0 1 . 0 1
0 0 1 0 0 0

1 1 0 1 0 1
4. Show that the zero and the unity are unique in a field.

5. Do the sets N (of natural numbers) and Z (set of integers) form fields? Justify your answers.
Also verify that the set C of complex numbers is a field.
6. Show that the field C of Complex numbers is not an ordered field.

7. (i) Define a set which is bounded below. Also define a lower bound of a set.
(ii) Give at least two examples of a set (one of an infinite set) which is bounded below
and mention a lower bound in each case.

(iii) Is the set of negative integers bounded below? Justify your answers.
8. Test which of the following sets are bounded above, bounded below, bounded and
unbounded.

(i) The intervals ]a, b], [a, b], ]a, b] and [a, b[, where a and b are any two real numbers.
(ii) The intervals [2, [, ]–, 3[, ]5,  [ and ] –, 4].
(iii) The set {cos e, cos 2 , cos 3 e, ......}.
(iv) S = {x  R : – a  x  a} for some a R.

Answers: Self Assessment

1. f(n) = 2n 2. f(n) = n + 1

3. infinite subset 4. countable
5. positive rational

2.7 Further Readings

Books Walter Rudin: Principles of Mathematical Analysis (3rd edition), Ch. 2, Ch. 3.
(3.1-3.12), Ch. 6 (6.1 - 6.22), Ch.7(7.1 - 7.27), Ch. 8 (8.1- 8.5, 8.17 - 8.22).
G.F. Simmons: Introduction to Topology and Modern Analysis, Ch. 2(9-13),
Appendix 1, p. 337-338.

Shanti Narayan: A Course of Mathematical Analysis, 4.81-4.86, 9.1-9.9, Ch.10,Ch.14,
Ch.15(15.2, 15.3, 15.4)
T.M. Apostol: Mathematical Analysis, (2nd Edition) 7.30 and 7.31.

S.C. Malik: Mathematical Analysis.
H.L. Royden: Real Analysis, Ch. 3, 4.

42 LOVELY PROFESSIONAL UNIVERSITY