Consider the set of strings. Prove by Induction

Consider the set of strings L S (0,1)* defined recursively as follows: The string 1 is in L For any string x in L, the string is also in L. For any string x in L, the string x0 is also in L For any strings x and y in L, the string xly is also in L These are the only strings in L (a) Prove by induction that every string w E L contains an odd number of 1s. (b) Is every string w that contains an odd number of 1s in L? In either case prove your answer. Let #(a, w) denote the number of times symbol a appears in string w; for example, #(0, 101110101101011)-5 and #(1, 101110101101011) = 10. You may assume without proof that # (a, uv)-#(a, u) + #(a, v) for any symbol a and any strings u and v, or any other result proved in class, in lab, or in the lecture notes. Otherwise, your proofs must be formal and self-contained