Find minimal dfa's for the following languages. In each case, prove that the result is minimal.

a$.L=\{a^n{b}^m$:$n2,m1\}.$

b$.L=\{a^{n}b$:$n1\}\{b^{n}a$:$n1\}.$

c$.L=\{a^n$:$n0,n2\}.$

d$.L=\{\begin{array}{c}{a}^n\text{:}n3\end{array}$ and $\{$:$n4\}.$

e$.L=\{a^n$:nmod$3=1\}\{a^n$:nmod$5=1\}.$