Q points$).$ In this problem, we will classify all regular languages over the unary

alphabet $=\{1\}.$

$($a$)$ Let $a,bin{Z}_{?}0.$ Show that the following language is regular:

$L=\{1^{ax+b}$:$xin{Z}_{?}0\}$

$($b$)$ Let $a_1,b_1,$dots,$a_k,b_{k}in{Z}_{?}0.$ Show that the following language is regular:

$L=\{1^{a_{i}x+b_i}$:$xin{Z}_{?}0,$iin$\{1,$dots,$k\}\}$

$($c$)$ Show that every regular language $L$ over the unary alphabet is of the form

$L=\{1^{a_{i}x+b_i}$:$xin{Z}_{?}0,$iin$\{1,$dots,$k\}\}$

for some $a_1,b_1,$dots,$a_k,b_{k}in{Z}_{?}0.$