# pumping lemma – Prove that \$0^m1^n\$ is not regular with \$m neq n\$

I need to show that $${0^m1^n} text{ with } m neq n$$ is not regular.

I tried to do this using the pumping lemma. Take $$s=0^{d_1} 1^{d_2}$$ with $$d_1 neq d_2$$. Given a decomposition $$s=xyz$$, there are $$3$$ possibilities.

$$1.$$ $$y$$ contains only $$0$$‘s.

If $$d_1 < d_2$$ and $$|y|=1$$, $$d_1$$ will eventually be equal to $$d_2$$, which is not accepted by $$L$$.

$$2.$$ $$y$$ contains only $$1$$‘s.

If $$d_2 < d_1$$ and $$|y|=1$$, $$d_1$$ will eventually be equal to $$d_2$$, which is not accepted by $$L$$.

$$3.$$ $$y$$ has the form $$(0)^+(1)^+$$

We illustrate this with an example. If we take $$xyz$$, then $$xyyz$$ will have a $$1$$ flowed inside the row’s of $$0$$‘s and that is not accepted by $$L$$ as first we have all $$0$$‘s and then $$1$$‘s.

Consequently, $$L$$ cannot be regular.