Answered: Suppose that A is not an empty set, and…

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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The topic is boundedness and fields in calculus.
1. Suppose that A is not an empty set, and is a set of real numbers bounded below. Let -A be the set of all numbers -x with x ∈ A.
Prove that infA = -sup(-A).
PS: inf means least upper bound and sup means the greatest lower bound.
2. Suppose that 0 < s ∈ R and fix s. Put a,b,c,d ∈ Q where b,d > 0 and f = a/b = c/d such that s^r = (s^a )(1/b).
Now if x ∈ R, define S(x) = {S^t : t ∈ Q and t less than or equal x}

Prove that S^t = supS(f)
PS: sup is the same as the greatest lower bound
3. Prove that if the multiplicative axiom of a field implies that if x≠0 and xy = x, then y =1.

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