Question

Problem 8.43. For sake of a contradiction, assume the interval (0,1) is countable. Then there exists a bijection f : N-> (0,1). For each n є N, its image under f is some number in (0, 1). Let f(n) :-0.aina2na3n , where ain 1s the first digit in the decimal form for the image of n, a2 is the second digit, and so on. If f (n) terminates after k digits, then our convention will be to continue the decimal form with 0's. Now, define b - 0.bib2b3..., where 2, ifAi#2 3, if a 2 (a) Prove that the decimal expansion that defines b above is in standard form. 3Hint: A countable union is a union of countably many sets. Recall that a countable set may be finite or infinite. Consider three cases: (1) finite union of countable sets (use induction with base casen 2), (2) countably infinite union of finite sets, (3) countably infinite union of countably infinite sets. CHAPTER 8. CARDINALITY (b) Prove that for all n E N,f(n)-b. (c) Prove that f is not onto. (d) Explain why we have a contradiction. (e) Explain why it follows that the open interval (0,1) cannot be countable.

all parts A-E please.

Answer 1

whese cem defned

no preimage e 兆etween