Assume that every point x E [0,1] has a binary expansion and a ternary expansion; i.e., there exist ak E {O, I} and bk E {O, 1, 2} such that

(For example, if x = 1/3, then a2k-1 = 0, a2k = 1 for all k and either b1 = 1, bk = ° for k > 1 or b1 = ° and bk = 1 for all k > 1.)

(a) Prove that E is a nonempty compact set of measure zero.

(b) Show that a point x E [0,1] belongs to E if and only if x has a ternary expansion whose digits satisfy bk -I- 1 for all kEN.

(c) Define f : E --4 [0,1] by f (~ ~~ ) = ~ b~~2 .