Recursion and structural induction $($a$)$ Consider $,$ an alphabet, and $,$ the set of strings over $.$ Assume that: $$ in $$w in $,$ c in $($wc in $)$ Complete the recursive definition for d : $,$ where d$($w$)$ is a string where each of w$$s characters is duplicated $($e$.$g$.,$ d$($LEAF$)=$ LLEEAAFF$).$ Basis: Consider the empty string, $.$ Write the basis step of the definition of d for this case. Recursive step: Consider a string that contains at least one character. Write the recursive step of the definition of d for this case.