Let $A=\{a,b\}.$

$($a$)$ Give a recursive definition for $A^{*}.$

$($b$)$ The set $A^{+}$is the set of strings over the alphabet $\{a,b\}$ of length at least $1.$ That is $A^{+}=A^{*}-\{\}.$ Give a recursive

definition for $A^{+}.$

$($c$)$ Let $S$ be the set of all strings from $A^{*}$ in which there is no $b$ before an $a.$ For example, the strings $,aa,bbb,$ and aabbbb

all belong to $S,$ but aabab!inS. Give a recursive definition for the set $S.($Hint: a recursive rule can concatenate

characters at the beginning or the end of a string.$)$

$($d$)$ For $xin{A}^{*},$ let bCount $(x)$ be the number of occurrences of the character $b$ in $x.$ Give a recursive definition for

bCount.