Chapter 5.9, Problem 9ES

Define a set S of strings over the set {a, b} recursively as follows:

I. Base: $\lambda \in S$

II. Recursion: If $s\in S$ . then
II(a) $bs\in S$II(b) $sb\in S$II(c) $saa\in S$
II(d) $aas\in S$

III. Restriction: Nothing is in S other than objects defined in I and II above.
Use structural induction to prove that every string in S contains an even number of a’s.