P2.5.10 It is possible for two different finite languages X and Y to have the same Kleene star, that is, for X* = Y* to be true. (a) Prove that X* = Y* if and only if both X CY* and Y CX*. (b) Use part (a) to show that X* = Y* if X = {a, abb, bb} and Y = {a, bb, bba}. (c) Prove that if X* = Y* and 1 & XUY, then the shortest string in X and the shortest string in Y have the same length. P2.5.10 It is possible for two different finite languages X and Y to have the same Kleene star, that is, for X* = Y* to be true. (a) Prove that X* = Y* if and only if both X CY* and Y CX*. (b) Use part (a) to show that X* = Y* if X = {a, abb, bb} and Y = {a, bb, bba}. (c) Prove that if X* = Y* and 1 & XUY, then the shortest string in X and the shortest string in Y have the same length