Question

2. Determine whether the given sets are countable or uncountable. Justify each answer with a bijection (or table like we did with Q+) or using results from class/textbook. (a) {0, 1, 2} * N (b) A = {(x, y) : x2 + y2 = 1} (c) {0, 1} R Che set of all 2-element subsets of N (e) Real numbers with decimal representations consists of all 1s. (f) The set of all functions from {0,1} to N

Answer 1

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(^) Suice, {0, 1, 2 3 and N Jbeth are cannitable , {0,1,23 x N in camntable. Cb) defrined ty: :(0, 2n] -→ A fex)= C cos x, sinx) -V n.? bijection : gies A vù aha mcanntable Simce, co, 2 R) in uncanntab le. 80, 13 xR m (e) Snce, R in umeamntable, uneauntable. (d) Pet A te thhe vet of all 2- dement mbreti ob N. ghem, f: A - NX N guen by'? fC{a,63) = (a,b) vinjection Since, NxN in rountable, no A in aho rsmtable.

I where A in the A f: [0, 1] defiied by: f(co.miMs? H, 123- -. Xn-- (0.,12 X3-.. Un---)10 E C 2 i Jone 2 and C-)10 in bare 10] injection E Provided ave take only giver one binary neforesentation fer each n win co,1]] [o, 1] in mncanntable, A is also uneartoble. Simie, fndhon fram (o, 1) to N. com abos be defined by (f(1, fc2). define a funetrón : ($) Let f be Now, Joy: P: A N X N bijedion then, $ grie. in conntable, A in aba fiice, N x N canntable