Answered: Z ∪ [1, +∞)

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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Question

(a) Design a bijection between Z ∪ [1, +∞) and (0, +∞). Justify your answer.
(b) Consider the infinite set S and a countable set A disjoint from S. Design a bijection between A ∪ S and S. (Hint: how is Theorem 10.3.26: (If S is an infinite set, then ℵ0 ≤ |S|.

Proof of theorem 10.3.26. To establish this, we must show that S has a subset S0 of cardinality ℵ0.

We proceed as follows. Since S is infinite, it surely contains some element, say s1.

Similarly, S \ {s1} (i.e., the set obtained from S by removing s1) contains some

element, say s2. Similarly, S \ {s1, s2} contains some element s3. Proceeding in

this manner creates an infinite sequence (s1, s2, s3, . . .) of elements of S. Let S0 =

{s1, s2, s3, . . .}. Then clearly |S0| = |N| = ℵ0. Since S0 is a subset of S, it follows

that ℵ0 ≤ |S|. Thus, ℵ0 is the smallest infinite cardinality) and part (a) are relevant to this question? Also you can recycle ideas and proofs from part (a).)

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