# 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.

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*}

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martini you are correct according to above example you have given even But in question mn which is product >= 10...As I think this will not going to be regular..if mn <= 10 then it will dfenitely going to be regular correct me if i am wrong – Nishant Nov 15 '12 at 14:25
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
Ok acording to you {a^nb^nc^n where n<= 10 } is regular then it's complement L={ a^nb^nc^n n>10 will be regular} is it fine as we know { a^nb^nc^n n>10 will be regular} is not regular.... – Nishant Nov 15 '12 at 14:35
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
martini: Is {aibj∣1≤i,1≤j,ij<10} it complement of my problem I don't think so Might be we are interpreting it wrong martini can you check again my above problem is it regular or not In my point of view not looking regular.. – Nishant Nov 15 '12 at 14:54