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Let L be any language. Define even(w) as the string obtained by extracting from w the letters in even-numbered positions; that is, if

w = a1 a2 a3 a4 ...

then

even(w) a2 a4 ...

Corresponding to this, we can define a language

even(L) = {even(w) : w $\epsilon$L}.

Prove that if L is regular, so is even(L).

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