Regular Language : $\{a^m b^n \mid mn \ge 10\}$

Question

I am little bit confuse here about below language is it regular language $$ L= \{a^m b^n \mid mn \ge 10 \}. $$

Answer 1

As the language $$ L_1 = \{ a^ib^j \mid 1 \le i, 1\le j, \;ij < 10\} $$ is finite, it is regular. Moreover $$ L_2 = \{a^ib^j \mid i,j \ge 1 \} $$ is regular (in has $aa^*bb^*$ as regular expression). But so $L = L_2 \setminus L_1 = L_2 \cap \complement L_1$ as the set of regular languages is closed under taking complements and finite intersections.

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Edit: Regarding Nishant's questions. We have by the definition of complement and intersection \begin{align*} L &= \{a^n b^m \mid nm \ge 10\}\\ &= \{a^n b^m \mid \neg(nm < 10)\}\\ &= \{w \in \Sigma^* \mid \exists n,m\ge 1 : w = a^nb^m \wedge \neg(nm < 10)\}\\ &= \{w \in \Sigma^* \mid \exists n,m \ge 1 : w = a^nb^m\} \cap \{w \in \Sigma^* \mid \neg\exists n,m \ge 1: w=a^nb^m, nm < 10\}\\ &= L_2 \cap \complement L_1 \end{align*}