1. (10 points) Regular expressions are built up using three operations: union (U), concatenation (o or placing regular expressions next to one another), and Kleene star ( *). In this question, you'll explore whether some of these are superfluous. Recall that regular expressions R1 and R2 are called equivalent if the languages they describe are equal sets, that is if L(Ri) - L(R2) For this question, the alphabet is {a, b}. a. Can the regular expression (a Ub)* be rewritten as an equivalent regular expression that does not include any union operations? If so, give that equivalent regular expression; if not, briefly justify why not b. Can the regular expression (aUb) be rewritten as an equivalent regular expression that does not include any union operations? If so, give that equivalent regular expression; if not, briefly justify why not c. Can the regular expression (a Ub)* be rewritten as an equivalent regular expression that does not include any Kleene star operations? If so, give that equivalent regular expression; if not, briefly justify why not. d. Can the regular expression (e*) be rewritten as an equivalent regular expression that does not include any Kleene star operations? If so, give that equivalent regular expression; if not, briefly justify why not Ectension (not for credit - do not hand in): Make a conjecture for a characterization of those languages that can be described by regular ea pressions that do not include ??? Kleene star operations. Similarly, make a conjecture for a characterization of those languages that can be described by regular expressions that do not include any union operations. Can you prove your conjectures