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

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

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Some body will edit my question I am not able to write it properly in mathematics form –  Nishant Nov 15 '12 at 13:44

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.
But, as I wrote, your $L$ (product $\ge 10$) is the complement of my $L_1$ (product $<10$) intersected with $aa^*bb^*$, and hence, regular. –  martini Nov 15 '12 at 14:28
No, as this is not the complement. The complement in $a^+b^+c^+$ is $L = \{a^nb^kc^l \mid n\ne k \vee n \ne l \vee k \ne l \vee n > 10\}$ –  martini Nov 15 '12 at 14:40