Characterizing the strings in a recursively defined set.
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)
(a) use structural induction to prove that if a string x ∈ s, then x does not have two or more consecutive a's.
(b) use strong induction on the length of a string x to show that if x does not have two or more consecutive a's, then x ∈ s. specifically, prove the following statement parameterized by n:

for any n ≥ 0, let x be a string of length n over the alphabet {a, b} that does not have two or more consecutive a's, then x ∈ s.