Question

state and prove fundamental theorem on equivalence relation

Answer 1

Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set. Let G denote the set of bijective functions over A that preserve the partition structure of A: ∀x ∈ A ∀g ∈ G (g(x) ∈ [x]). ... (This is the Fundamental Theorem of Equivalence Relations, mentioned above);