Fix an alphabet σ. we define the notion of an awesome language recursively as follows. • ∅ is awesome; • {a} for each a ∈ σ is awesome; • if l1 and l2 are awesome, then l1 ∪ l2 is awesome; • if l1 and l2 are awesome, then l1l2 is awesome; so an awesome language is any language that can be constructed by starting from the base cases and applying the recursive rules a finite number of times (hopefully it is clear which ones above correspond to the base cases and which ones correspond to the recursive rules). we define the notion of a super awesome language recursively as follows. • ∅ is super awesome; 2 • {a} for each a ∈ σ is super awesome; • if l1 and l2 are super awesome, then l1 ∪ l2 is super awesome; • if l1 and l2 are super awesome, then l1l2 is super awesome; • if l is super awesome, then l∗ is super awesome. 1. give a simple characterization for the set of all awesome languages. briefly justify your answer. a detailed argument is not needed. 2. is the set of all awesome languages the same as the set of all super awesome languages? briefly justify your answer using part (a). 3. for a function f : σ → σ∗ and a language l ⊆ σ∗, define the language lf = {f(a1)f(a2)···f(ak) : for all i, ai ∈ σ, and a1a2 ∈ l}. prove that for any f : σ → σ∗, if l is super awesome, then so is lf .