Let M = (Q, , q0, A, ) be an FA. Below are other conceivable methods of defining the extended transition function (see Definition 2.12). In each case, determine whether it is in fact a valid definition of a function on the set Q , and why. If it is, show using mathematical induction that it defines the same function that Definition 2.12 does.

a)For every q Q, (q, )=q; for every y , , and q Q, (q, y ) = ((q, y), ).

b)For every q Q, (q, )=q; for every y , , and q Q, (q,y) = ((q,),y).

c)For every q Q, (q, )=q; for every q Q and every , (q,) = (q,); for every q Q, and every x and y in , (q, xy) = ((q, x), y).