I'd really love your help with the following:
For any fixed $L_2$ I need to decide whether there is closure under the following operators:
$A_r(L)=\{x \mid \exists y \in L_2 : xy \in L\}$
$A_l(L)=\{x \mid \exists y \in L : xy \in L_2\}$.
The relevant options are:
Regular languages are closed under $A_l$ resp. $A_r$, for any language $L_2$
For some languages $L_2$, regular languages are closed under $A_l$ resp. $A_r$, and for some languages $L_2$, regular languages are not closed under $A_l$ resp. $A_r$.
I believed that the answer for (1) should be (2), because when I get a word in $w \in L$ and $w=xy$ I can build an automaton that can guess where $x$ turning to $y$, but then it needs to verify that $y$ belongs to $L_2$ and if it won't be regular, how would it do that?
The answer for that is (1).
What should I do in order to analyze those operators correctly and to determine if the regular languages are closed under them or not?