(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)

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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 1 (56.63 KiB) Viewed 9 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]