Exercise 9 (Ex. 44, Chapter 1 of [Martin; 2011]). Each case below gives a recursive definition of a subset L of {a,b}. Give a simple non-recursive definition of L in each case. a. aL; for any xL,xa and xb are in L. b. aL; for any xL,bx and xb are in L. c. aL; for any xL,ax and xb are in L. d. aL; for any xL,xb,xa, and bx are in L. e. aL; for any xL,xb,ax, and bx are in L. f. aL; for any xL,xb and xba are in L. Hint: The non-recursive definition of the language listed at point a may be given as: "the set of all strings beginning with a." Exercise 10 (Ex. 50, Chapter 1 of [Martin; 2011]). Prove using mathematical induction that for every positive integer n, i=1ni2i=(n1)2n+1+2