1.8.6. The star height h(a) of a regular expression a is defined by induction as follows.

The star height h (alpha) of a regular expression a is defined by induction as follows. h () = 0 h (a) = 0 for each a element sigma h (alpha Union beta) = h (alpha beta) = the maximum of h (alpha) and h (beta). h (alpha*) = h (alpha) + 1 For example, if alpha = (((ab) * Union b*)*Ua*), then h (alpha) = 2. Find, in each case, a regular expression which represents the same language and has star height as small as possible. (a) ((abc)*ab)* (b) (a(ab*c)*)* (c) (c(a*b)*)* (d) (a*Union b*Union ab)*