Assume that you know that ~ is an equivalence relation on R\ {0} defined as follows: xy > 0. Find all the distinct equivalence classes of ~.

Let B be the set of equivalence classes for R. Show that the function f : B → Z, given by f([(a, b)]) = a − b, is well-defined and bijective. Please really show it step by step, show the steps you need to take and explain them in detail, thank you in advance

18. Let R be the relation on R defined by xRy → x – y is an integer. Prove that R is an equivalence relation.

3 For the equivalence relation on {0, 1,2,3,4} defined as {(x, y) : x +y is even} (a) Write it out as a list of ordered pairs (b) List the cells in the associated partition.