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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.

John Doe
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