I am stuck on the following question.If $L$ is regular show that $\mathrm{even}(L)$ is also regular, where $\mathrm{even}(L) = \{ even(w) : w \in L \}$, $w$ is a string in $L$ and $\mathrm{even}(w)$ is the string obtained by extracting from $w$ the letters in even numbered positions.How can I prove that $\mathrm{even}(L)$ is also regular?
If L is regular, show that even(L) is also regular
formal languages;regular languages;closure properties
We can also solve this question using closure operations. Let $\Sigma$ be the original alphabet, and let $\Sigma' = \{x' : x \in \Sigma\}$ be a second copy of the alphabet. Define two homomorphisms $h$ and $d$ by $h(x) = h(x') = x$ for all $x \in \Sigma$ and $d(x) = x$, $d(x') = \epsilon$ for all $x \in \Sigma$. Then$$\operatorname{even}(L) = d(h^{-1}(L) \cap (\Sigma'\Sigma)^*(\epsilon+\Sigma')).$$