Are these languages Regular or Non-regular?

Question

I have these two languages

$L={\{a^n b^m,n≥m+5,m>0}\}$ Where $∑=(a,b)$

$L={\{a^n b^m,n≥m+5,m≤5}\}$ Where $∑=(a,b)$

As you can see that there is only one difference, the condition of m is different.

What my question is that, are these languages regular or non-regular?

If these both languages are non-regular then how can we able to prove that using pumping lemma?

Answer 1

Hint:

• Split $a^n$ into $\color{blue}{a^n}$ and $\color{green}{a^5}$, that is, consider language $$L = \{\color{blue}{a^n}\color{green}{a^5}b^m \mid \color{blue}{n}\geq m, m > 0\},$$ where the two new letters $\color{blue}{a}$ and $\color{green}{a}$ are just two distinguishable letters $a$, and $\color{blue}{n} = n-5$.
• For the second language observe that \begin{align}\{a^nb^m\mid n\geq m+5, m\leq 5\} &= \bigcup_{i=0}^{5}\ \{a^nb^m\mid n\geq m+5, m=i\} \\ &=\bigcup_{i=0}^{5}\ \{a^nb^i\mid n\geq i+5\}\\ &=\{a^{\color{blue}{n}}a^5b^0\mid \color{blue}{n}\geq 0\} \cup \ldots \cup \{a^{\color{blue}{n}}a^{10}b^5\mid \color{blue}{n}\geq 0\}.\end{align}

I hope this helps $\ddot\smile$