A partition P of Z is said to be compatible with + if given any X, Y ∈ P and any x1, x2 ∈ X, y1, y2 ∈ Y the elements x1 + y1 and x2 + y2 belong to the same member of P. Suppose that P is a partition of with 4 members compatible with +. Show that P is the partition associated to the equivalence relation congruence modulo 4. Can you see how to generalize this to a partition of Z compatible with + having an arbitrary amount of classes (not just 4).

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