Find minimal dfa's for the following languages. In each case prove that the result is minimal.
(a) $L =$ {$a^n b^m :n≥2,m≥1$}.
(b) $L =$ {$a^n b:n ≥0$} $∪$ {$b^n a:n ≥1$}
(c) $L =$ {$a^n :n ≥ 0,n ≠ 3$}.
(d) $L =$ {$a^n:n ≠ 2$ and $n ≠4$}.
(e) $L =$ {$a^n:n$ mod $3 = 0$} $∪$ {$a^n: n$ mod $5 = 1$}.