Suppose the language, L, over an alphabet Σ0 is regular, and f : Σ0 → Σ1, is a function from the alphabet Σ0 to the alphabet Σ1. The function f can be lifted to words producing the function: ˆf : Σ∗0 → Σ∗1 ˆf(ε) = ε ˆf(aw) = f(a)ˆf(w) Show that the language Lf = {ˆf(w) ∈ Σ∗1| w ∈ Σ∗0} is regular. Hint: You have to use the formal definition of DFAs.

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