Question: Let and be alphabets with . Let M be a DFA with alphabet.1. Prove that there exists a DFA Mwith alphabet such that L(M) =L(M).

Let and be alphabets with . Let M be a DFA with alphabet.1. Prove that there exists a DFA Mwith alphabet such that L(M) =L(M). (Hint: add afail 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 ,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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