The recursive definition given below defines a set S of strings over the alphabet {a, b}: Base case: λ ∈ S and a ∈ S Recursive rule: if x ∈ S then, xb ∈ S (Rule 1) xba ∈ S (Rule 2) This problem asks you to prove that the set S is exactly the set of strings over {a, b} which do not contain two or more consecutive a's. In other words, you will prove that x ∈ S if and only if x does not contain two consecutive a's. The two directions of the "if and only if" are proven separately. (a) Use structural induction to prove that if a string x ∈ S, then x does not have two or more consecutive a's.