$1.$

$(8$ points$)$ Define a set S of strings over integers recursively as follows:

$1.$ Base: I S$,3$ e S$,5$ S$,7$ e S$,9$ S

Il$.$ Recursion: If s e S and t e S$,$ then

Il$($a$)$ st e s

Il$($b$)2$s S

Il$($c$)4$ s G S

Il$($d$)$ e S

Ill. Restriction: Nothing is in S other than objects defined in I and Il$.$

Use structural induction to prove that every string in S represents an odd integer when written

in decimal notation.