prove using induction please

(a) A set of strings L (over alphabet sigma = {0, 1}) is defined recursively as follows: the empty string epsilon elementof L, and for all strings x, y elementof L, 1x0y1 elementof L. Answer the following questions: i. Let #_a(w) be the number of occurrences of the character a in w. Formally prove that every string w elementof L has #_1 (w) = 2 times #_0 (w). In other words, every string has exactly twice as many 1s as 0s. ii. Prove that there is a string w elementof {0, 1}* with #_1 (w) = 2 times #_0(w) and w elementof L. (b) Let w be a string consisting of entirely 0's and 1's (i.e., w elementof {0, 1}*). The complement of w, w^c, is obtained from w by changing every 0 in w to a 1 in w^c, and every 1 in w to a 0 in w^c. Give a formal and precise definition of w^c, and formally prove that (xy)^c = x^c y^c for all strings x, y