i have found the equivalence classes of given $R_L$ and i need to find the separating words between the equivalence classes(which i don't know how to do). would appreciate if you could explain to me how to do so. also, if you can, would appreciate if you could check if i obtained the correct equivalence classes for both languages.
1)$L=\bigg\{w\in \sum^*\bigg| w\quad \text{starts and ends with} \quad aa\bigg\}$
equivalence classes i've obtained:
$S_1 = \epsilon$
$S_2 = a$
$S_3 = (b+c)\sum^*+a(b+c)\sum^*$
$S_4 = aa+aaa+aa\sum^*aa$
$S_5 = aa\sum^*(b+c)a$
$S_6 = aa\sum^*(b+c)$
2)$L=\{\sum^*-(\{\epsilon, a,b\}\cup \{bba^i|i\ge 0\})\}$
equivalence classes i've obtained:
after joining $\{\epsilon, a,b\}\cup \{bba^i|i\ge 0\}=\{\epsilon, a,b,bba^*\}$ and using the complementary, i've obtained:
$S_1 = \epsilon$
$S_2 = a$
$S_3 = b$
$S_4 = bba*$
$S_5 = c\Sigma^*+a\Sigma^++b(a+c)\Sigma^*+bb\Sigma^*(b+c)\Sigma^*$
how can i find the separating words? also, if you can, could you verify that i've obtained the correct results?
thank you very much!