I have the language $ L$ $ =$ {$ a$ $ =$ $ b$ $ +$ $ c$ $ |$ $ a$ , $ b$ , $ c$ are binary integers, and $ a$ is the sum of $ b$ and $ c$ }, with alphabet $ \sum =\left\{0,\:1,\:+,\:=\right\}$ . I need to prove that this language is not regular using the Myhill-Nerode Theorem.

I am able to prove it using the pumping lemma by using $ a$ $ =$ $ 111$ $ =$ $ b$ $ +$ $ c$ $ =$ $ 100$ $ +$ $ 11$ . This string has a pumping length of 2. I divide the string into $ x$ $ =$ $ 1$ , $ y$ $ =$ $ 1$ and $ z$ $ =$ 1=100+11. So if I pump $ y$ with $ i$ $ =$ $ 2$ , I change $ y$ to $ 11$ . Therefore, the string after pumping is $ 1111=100+11$ , which is not true, leading to a contradiction that violates the pumping lemma.

But I am not sure how I would approach this using the Myhill-Nerode theorem. I know the idea is to find an infinite set in $ L$ that is pairwise distinguishable. But I am not sure which set I could choose. Could I use the same sets as the ones I used with the pumping lemma or would it have to be something different?

Thank you!