Question

Exercise 7.H.

7.Н. Show that every number in the Cantor set has a ternary (-base 3) expan- sion using only the digits 0, 2 7.I. Show that the collection of "right hand" end points in F is denumerable. Show that if all these end points are deleted from F, then what remains can be put onto one-one correspondence with all of [0, 1). Conclude that the set F is not

Answer 1

"We now claim that the cantor set consist precisely of numbers of the form x = sigma^infinity_k = 1 a_k/3^k where each a_k is either 0 or 2 the map {0, 2}^N rightarrow c is then a bijection by the above observation suppose x is given by then 1/3 x = sigma^infinity_k = 1 b_k/3^k where b_1 = 0, b_k = a-k, it k > 2 1/3 x + 2/3 = sigma^infinity_k = 1 b_k/g^k where b_1 = 2, b""_k = a_k if k > 2 thus x elementof c_1 if and only if it equals a ternary expansion where either a_1 = 0 or a_2 = 2 since it is of the form 1/3 y or 2/3 + 1/3 y for some y elementof [0, 1] Repeating the argument we have see that x elementof c_n iff it equals a ternary expansion where a_k i_2 either 0 or 2 for 1 < k & n It follows that the ternary"