Using closure properties to show that $L_1=\{a^lb^mc^m|l,m\ge 0\} \cup L(b^*c^*)$ is regular or not i'm trying to figure out whether this Union $\left [  L_1=\{a^lb^mc^m|l,m\ge 0\} \cup L(b^*c^*)\right]=K$ is regular or not, now since regular languages are closed under intersection, so i assume $K$ is regular then its intersection with $L(a^*b^*)$ should be regular which is $K\cap L(a^*b^*)=L(a^*)\cup L(b^*)$ right ?

1 - Does this implies that the Union is also regular ?

2 - i know that $L=\{a^lb^mc^m|l,m\ge1\}$ is CFL, but is $L_1$ also CFL ?, which is basically  $L_1=L\cup \{\epsilon ,a,bc\}$ 

3 - is $K$ regular or not and how to prove it ?

Any Hints would be appreciated.
You are not starting with the right proof idea. It's this:


I want to show $L$ is not regular.
Assume towards a contradiction that $L$ is regular.
Since REG is closed against $f$, I conclude that $L' = f(L)$ is also regular.
But I already know that $L'$ is not regular (from elsewhere). Contradiction; $L$ can't be regular.


See here for complete examples.