I have a couple of proeblems which I am not sure how to solve. I know that an equivalence relation is a relation set that corresponds to the properties: reflexive, symmetric, anti-symmetric and transitive.
1) Consider the alphabet Σ = {a,b}. For which languages L does the equivalence relation RI have exactly one equivalence class?
2) Let L be a language (not necessarily regular) over an alphabet. Show that if the equivalence class containing the empty string [ε] is not {ε}, then it is infinite.
3) Consider the language L over an alphabet Σ = {a,b} described as L = {x ∈ Σ: |na(x) = nb(x)}. Recall na(x) = number of a'sin x.
(1) Show that if na(x) - nb(x) = na(y) - nb(y), then xRIy.
(2) Show that if na(x) - nb(x) not = na(y) - nb(y), then x and y are L-distinguishable.
(3) Describe all the equivalence classes of RI.