A subset A of X is called an ”equivalence class” of ∼ if for all a1, a2 ∈ A we have that a1 ∼ a2, but also that for all

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A subset A of X is called an ”equivalence class” of ∼ if for alla1, a2 ∈ A we have that a1 ∼ a2, but also that for all a ∈ A and b∈ (X \ A), a and b are not equivalent.
(a): Define a relation on the integers such that aRb when a − bis even. Prove that this is an equivalence relation.
(b): Let A be any set and consider a function f from A to A.Define a relation such that a1Ra2 when f(a) = f(b). Prove that thisis an equivalence relation.
(c): What are the equivalence classes in the above examples?
(d): Is the relation xRy when |x − y| < 2 an equivalencerelation?
(e): Given an equivalence relation on X, can an element of X bea member of more than one equivalence class?