Part 2 builds from Part 1, so I'm posting them together.

Exercise 3. (5 points) Let and , be alphabets with c . Let M be a DFA with alphabet 1. Prove that there exists a DFA M, with alphabet , such that L(M') "fail" state to M.) Be sure to prove that your construction is correct. L(M). (Hint: add a 2. Let F be a binary operation applicable to any two formal languages. In particular, F may be applied to two languages over different alphabets. Now assume that for all alphabets , and for all regular languages A and B over , FA, B) is regular. Use Part 1 to conclude that for any two regular languages A and B (over possibly different alphabets), F(A, B) is regular