Page 27 - DMTH401_REAL ANALYSIS

Unit 1: Sets and Numbers

1.8 Review Questions Notes

1. Write the following in the set-builder form:
A = {2, 4, 6, ...}
A = {1, 3, 5, ...}
2. Write the following in the tabular form:

A = {x:x is a factor of 15}
A = {x:x is a natural number between 20 and 30}
A = {x:x is a negative integer}
3. Let X be a universal set and let S be a subset of X. Prove that
(i) P(0) = {}
c c
(ii) (S ) = S.
4. Let A, B and C be any three sets. Then prove the following:
(i) A  B = B  A,A  B = B  A (Commutative laws).

(ii) A  (B  C) = (A  B)  C, A  (B  C) = (A  B)  C (Associative laws).

(iii) A (B  C) = (A  B)  (A  C)
A  (B  C) = (A  B)  (A  C) (Distributive laws).
C
C
C
C
C
C
(iv) (A  B) =A  B , (A  B) = A  B (DeMorgan laws).
5. Justify that
(i) N is a proper subset of Z.
(ii) Z is a proper subset of Q.
Answers: Self Assessment
1. (a) 2. (b)
3. (c) 4. (c)

5. (c)
1.9 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.

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