(1) Find the upper and lower bound of the following sets also state if the set is bounded or not (i) A = -{} (ii) B = {-„} (iii) C = {2n} (iv) D = (0, 00) n+3 (2) Find the supremum and infimum of the sets in question (1) above. Also find out if the supremum or infimum belong to the set or not (3) If A = {1+ "} , find out if it is bounded or not. If it is then find the upper bound , lower bound , supremum and infimu of A.

(1) Find the upper and lower bound of the following sets also state if the set is bounded or not (i) A = -{} (ii) B = {-„} (iii) C = {2n} (iv) D = (0, 00) n+3 (2) Find the supremum and infimum of the sets in question (1) above. Also find out if the supremum or infimum belong to the set or not (3) If A = {1+ "} , find out if it is bounded or not. If it is then find the upper bound , lower bound , supremum and infimu of A.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

100%

(1) Find the upper and lower bound of the following sets also state if the set is bounded or not
(i) A = -{}
(ii) B = {-}
(iii) C = {2n1}
%3D
n+3
(iv) D = (0, ∞0)
(2) Find the supremum and infimum of the sets in question (1) above. Also find out if the supremum
or infimum belong to the set or not
(3) If A = {1+ D"} , find out if it is bounded or not. If it is then find the upper bound ,
lower bound , supremum and infimu of A.
(4) Let A CR be a et ith greatet element a E A Prove that sup(A)=a
(5) Let A = {r € R+ : 2² < 2} . Explain hy A i bounded above and below and find sup(A) and inf(A).
(6)Suppose A, B CR are both non-empty subsets with a = Sup(A) and 3 = sup(B) both exist-
ing. Then if A C B then a <B.
(7) Let A = {4, 7, 8} and B = {-1,5}, Find and compare
sup(A+B) , inf(A+B), sup(A)+sup(B),sup A - inf B, inf A-inf B
(8) Prove by an example that for any set A, g.l.b(-A) = –l.u.b(A)
(9) Prove and give an example that if A and B are two bounded non-empty subset of R, then
AUB is also bounded
(10) Given that A = {-21,20, 23} and B = {1, 10, 100}, Find
(i)Sup(AU B)
(ii) inf(AU B)
(iii)min.(inf.A, Inf.B)
(iv) таz.(sup.A, sup.B)
(11) Given that A = {-12, –11, 13} and B = {-10, 20, 30}, Find
(i) g.l.b(-A)
(i) -1.и.Ь(—А)
(iii)Inf.(B – A)
(iv) Sup.(В — A)
(v) Sup.A – Sup. B)

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