Exercise 3. (5 points) Let E and E' be alphabets with E C E' (subset). Let M be a DFA with alphabet E.

1. Prove that there exists a DFA M' with alphabet E' such that L(M') = L(M). (Hint: add a

fail" state to M.) Be sure to prove that your construction is correct.

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 E, F(A;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.