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.