(30 points) Bonus Given: If R is a regular expression that defines the language L(R), there is a regular expression S that defines the language L(S) = L(R), i.e. regular expressions are closed under the complement operation (the complement of a regular language is also a regular language). Note: Even though we will discuss them later in class, you are not allowed to use automata for the proofs in this question. Show that: (a) (10 points) The union of two regular languages is also regular: Given two regular expressions R and R2, define a regular expression S such that L(S) = L(Ri) U L(R2). (b) (20 points) The intersection of two regular languages is also regular: Given two regular expressions R and R2 show that there is a regular expression S such that L(S) = L(Ri) n L(R2). Note: You do not have to construct S, only show that it must exist, given the property mentioned above