Question: Let and ' be alphabets with '. Let M be a DFA with alphabet . 1. Prove that there exists a DFA M' with alphabet

Let and ' be alphabets with '. Let M be a DFA with alphabet . 1. Prove that there exists a DFA M' with alphabet ' such that L(M') = L(M). (pointer: use a "fail" state in M.) 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 , 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.

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