Question

2. (a) Prove by structural induction that for all x E {0,1}*, \x = x. (b) Consider the function reverse : {0,1}* + {0,1}* which reverses a binary string, e.g, reverse(01001) = 10010. Give an inductive definition for reverse. (Assume that we defined {0,1}* and concatenation of binary strings as we did in lecture.) (c) Using your inductive definition, prove that for all x, y E {0,1}*, reverse(xy) = reverse(y)reverse(x). (You may assume that concatenation is associative, i.e., for all x, y, z E {0, 1)*, x(yz) = (xy)z.)

Answer 1

→ The objective je prove by structural induction that for all xt {0,03* Ax=x. det 'm' be an orbitasy Stroma . Assume for any proper substring 'w" of "x" that I w.l=w) The above Substring rises for two cases:- If x=x then **= Adat [By definition of] Case in some symbol a. . If If one at is then a= a.w for empty card not then one ca.w).a za [w.17 [Definition of Inductor hypothesis] = a.w 12.8 = al Txokoa an both cases Hence, it is proved

2 -b). Consider which reverses the function the binary reverse : { 0, y = {0,13* string. Reverse (01001) = 10010 het ", y denotes conventation of 'n' and y' "au reversal the of Reverse denotes 2 ply. it > I alwy) it is, a.w s E if xəe (1) L reverse (co), ait 1-06 Reverse (reverse (W) *). By using inductive definition prove that for all rey € (0.) ; reverse (ny) = reversely) If y are orbitary Strings Assume for any proper substring of a that reverse (wy) - reverse ly). reverse wo) If there are two cases either a is empty a cor not.