# Finding the equivalence classes of a language

I'm doing a problem where I need to find the $$≡_A$$ equivalence classes of the language $$A = \{ 0^{n}x \mid n \in \mathbb Z^+, x \in \{0, 1\}^*, \text{ and } \#_0(x) ≥ n \}.$$

The best way I've learned to find the equivalence classes from a formal language is to create an automation and minimize it. Is there a conventional way of finding the equivalence classes of a language more quickly and intuitively? Because I do not know the DFA for this problem. Thanks for the help!

Your language is $$01^*0(0+1)^*$$, i.e., all words starting with 0 and containing at least one more 0.
Indeed, a word of the form $$0y0z$$ is in $$A$$ since we can take $$n=1$$ and $$x = y0z$$. Conversely, if $$w = 0^nx \in A$$ then $$w$$ contains at least $$2n \geq 2$$ many 0's, and starts with a 0.
Therefore the Myhill–Nerode equivalence classes of your language are: $$\epsilon, 1(0+1)^*, 01^*, 01^*0(0+1)^*.$$