A bomomorphism is a function $f$:$longrightarrow{}^{*}$ from one alphabet to strings over

another alphabet. We can extend $f$ to operate on strings by defining $f(w)=$

$f(w_1)f(w_2)cdotsf(w_n),$ where $w=w_1{w}_{2}cdots{w}_n$ and each $w_{i}in.$ 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 rec$-$

ognizes $B$ 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 in every case?

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

closed under homomorphism.