# Why do non regular languages have infinitely many equivalence classes?

For example, for the language $L = \{a^nb^m|n \neq m\}$, why does it have infinitely many equivalence classes? How do I show/see that?

• It might help if you state the equivalence you are referring to. – reinierpost Oct 22 '15 at 8:07

There are at least two ways. The first way is to give infinitely many pairwise inequivalent words. In the case of your language, you can take $\{ a^n : n \geq 0 \}$, for example (exercise).