Problem $1.$ Consider the following inductively defined set of $$good$$ strings of a$$s and b$$s:

$$ Seed: The empty string $\backslash$epsi is good.

$$ Rule $1$: If s is good, then asb is good.

$$ Rule $2$: If s is good, then bsa is good.

$$ Rule $3$: If s$1,$s$2$ are good, then s$1$s$2$ is good.

$($a$)$ Prove that abaababbba is good.

$($b$)$ Give a characterization of the good strings: i$.$e$.,$ a property P such that a string s is good if

and only if s has property P$.$

$($Your characterization should be non$-$trivial; e$.$g$.,$ P cannot be $$all good strings$.)$

$($c$)$ Prove that every good string satisfies your P$.($Soundness$.)$

$($d$)$ Prove that every string that satisfies your P is good. $($Completeness$.)$

$($e$)$ Prove that abababbba is not good.