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(b) Theorem: The set R is uncountable. Proof: Let f: N→ (0, 1) be an arbitrary function. (1) For each n € N, denote f(n)
Posted: Wed Jul 06, 2022 12:05 pm
by answerhappygod
- B Theorem The Set R Is Uncountable Proof Let F N 0 1 Be An Arbitrary Function 1 For Each N N Denote F N 1 (56.63 KiB) Viewed 11 times
(b) Theorem: The set R is uncountable. Proof: Let f: N→ (0, 1) be an arbitrary function. (1) For each n € N, denote f(n) by 0.an, ang ans,... where each an is a digit between 0 and 9 inclusive. (2) Define b € (0, 1) by b=0.b1b₂b3... where = {²} bk √2 if akk 3 if akk Now bf(n) for any n € N. Thus f is not an onto function. Therefore (0, 1) is an uncountable set. (i) Explain why (4) is true. (ii) Explain why (5) is true. (iii) Explain why (6) is true. (iv) Why does (6) prove the theorem? #2, = 2. (3) (4) (5) (6) [16 marks]