1k views

Let $L$ be any language. Define $\text{Even} (W)$ as the strings obtained by extracting from $W$ the letters in the even-numbered positions and $\text{Even}(L) = \{ \text{Even} (W) \mid W \in L\}.$ We define another language $\text{Chop} (L)$ by removing the two leftmost symbols of every string in $L$ given by $\text{Chop}(L) = \{W \mid \mathcal{v} W \in L,$ with $\mid \mathcal{v} \mid =2\}.$ If $L$ is regular language then

1. $\text{Even(L)}$ is regular and $\text{Chop(L)}$ is not regular
2. Both $\text{Even(L)}$ and $\text{Chop(L)}$ are regular
3. $\text{Even(L)}$ is not regular and $\text{Chop(L)}$ is regular
4. Both $\text{Even(L)}$ and $\text{Chop(L)}$ are not regular

edited | 1k views

+1 vote

Since in both case the word obtained after removal belongs to that particular language L. and the language L is regular

so both the Languages defined by even and Chop are regular