Let S be the set of all binary strings where every block of 1's have even length. [a] Determine an unambiguous expression for S. Explain your answer. [b] Determine the generating series of S with respect to the lengths of the strings. Simplify your answer as much as possible. [c] Let a\" he the number of strings in S of length it. Use part (b) to prove that for all n 2 2, an = (In1 + an2- [d] Give a combinatorial proof of the equation In part {c} uslng a hijectlon. For any bljection that you use, you need to {1) dene the domain and codornain; {2) define the function; [3) write down the inverse; and {4) show that both the function and the inverse are well-dened. You do not need to prove that your inverse is correct, but it should be clear that it reverses the operation of f