Is $\{a^mb^nc^n \mid m,n \geq 0\} \cup \{b,c\}^*$ regular or not? Show if $L = \{a^mb^nc^n \mid m,n \geq 0\} \cup \{b,c\}^*$ is regular or not.
My attempt:
I think the Pumping lemma won't work in that constellation, so I'm working with &quot;The intersection of regular languages is regular&quot;.
Assuming $L$ is regular. I already know that $S=\{ab^* c^*\}$ is regular. I'll now look at the intersection $L \cap S$.
$$(\{a^mb^nc^n \mid m,n \geq 0\}\cap \{ab^* c^*\}) \cup (\{b,c\}^*\cap \{ab^* c^*\})$$
$$\{ab^nc^n \mid n \geq 0\} \cup \emptyset = \{ab^nc^n \mid n \geq 0\}$$
Which is not regular according to Pumping lemma. Is this idea correct?
Pumping lemma works easily. Somewhere it says &quot;if the language is regular then for every string in the language...&quot; Just check the pumping lemma for strings that start with exactly one a.