Question

xercise Let D infinite het F := { X 1 X is a finite subset of D} Show i 0(D) = o(F) & fint Let Fly = { X CD) o(X) {m} olanlad show deo (Fin) Note that F = U FR Show X X X

We were given this exercise in class and I would greatly appreciate it if it was worked out.

Use the definitions for the hints given. I'm not sure what else to say.

Answer 1

obe infinite set. 032 sol So, two cases may arise, either dis Countable un Countable. oorder of o, o D) = X Caº Pha 7) if Dis Countable and, OCD) = c (continuum), it D is uncountable F = x is a finite subset of o of o} let, F = {x|xisa Singleton set ind 0} Now, o } Faz Ez مل (xlx = {a}, vacs Containty two elementy = {x1 x = 29,63, a, bED} { x 1 x has three elements} x=Labe}, abcco} and x 1 elements} { x 1 x = fan az 193, - - , an} 9, 9, 92=6, 9=C, and all@ceD} F = so on Fn- Contains no whene, so ano & an aiGD n- Since, each Fn Countable. son 2 In is the Countable cineon of Countable n=1 sets so, LOD E is Countably inferite So, onden of E te. (F)= W Xo So, if D is Countable set. Then. OCD) = O(A). but, th, is uncoumble. Then the statement is false.