Let A be a set, and suppose that ~ is a relation on A. (a) Define what it means to say that ~ is an equivalence relation on A. (b) Suppose that x E A. Give the definition of the equivalence class cl(x) of x with respect to the equivalence relation ~. (c) Using the properties of an equivalence relation (and explicitly stating them as you use them), prove that if ~y, then cl(x) = cl(y). (d) Write down an equivalence relation on the set of positive integers that has exactly four equivalence classes, of which two are infinite and two are finite.