A homomorphism is a function $f$:${}^{\text{'}}$ from one alphabet to strings over another

alphabet. We can extend $f$ to operate on strings by defining $f(w)=f(w_1)f(w_2)dotsf(w_n),$

where $w=w_1{w}_{2}dots{w}_k.$ and each $w_{t}in.$ We further extend $f$ to operate on languages by

defining $f(A)=\{f(w)|winA\},$ for any language $A.$

a$.$ Show, by giving a formal construction, that the class of regular languages is closed

under homomorphism. In other words, given a DFA M that recognizes $B$ and a

homomorphism $f,$ construct a finite automaton $M^{\text{'}}$ that recognizes $f(B).$ Consider the

machine $M^{\text{'}}$ that you constructed. Is it a DFA is every case?

b$.$ Show, by giving an example, that the class of non$-$regular languages is not closed

under homomorphism.

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