(a) Consider the recursively defined set of binary strings B defined by: Basis Step: 0B,1B Recursive Step: if xB then xxB where xx is the string concatenated with itself. Prove using structural induction that for all elements x of B, the length of x is an integer power of 2 . (b) Recall the recursive definition of N. Basis Step: 0N Recursive Step: If mN then m+1N. Consider the function sumeven :NN defined recursively as: Basis Step: sumeven (0)=0 Recursive Step: If mN then sumeven (m+1)=sumeven(m)+(2m+2) Use structural induction to show that for all nN, that sumeven (n)=n(n+1). (c) Consider the recursively defined set D of binary strings: Basis Step: 0D and 1D Recursive Step: If xD and wD then wxwD 1 Prove using structural induction that for all elements uD,u starts and ends with the same character