# How do I prove regular/non-regular with Nerode-Theorem? How to use it?

$$L_{1}=\left\{w \in\{a, b\}^{*} | \#_{a}(w)=0\right\}$$

$$L_{2}=\left\{w \in\{0,1\}^{*} | w=u v u \text { with } u, v \in\{0,1\}^{*}\right\}$$

I have problems to prove regularity with the nerode theorem

The idea behind this is that the nerode classes have to be finite for a regular language

How do I use this?

For L1 I know this

There is only $$b^{n}$$ due to the fact that there language does not accept any a

Let m$$\neq$$n than there exists $$b^{n}b$$ and $$b^{m}b$$ for all m $$\in \mathbb{N}$$\n an n $$\in \mathbb{N}$$\m and of all them are in L

For 2 I know this

Let vu be x so for all a,b $$\in \{0,1\}^{\star}$$ ax an bc are in L.

In this case, both are regular (if I understand the notation correctly, $$L_1$$ is just strings with no $$a$$; for $$L_2$$ you can take $$u = \epsilon$$ and $$v$$ the given string, so $$L_2 = \{1, 2\}^*$$). Nice red hering, BTW.