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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).

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).

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 3E: a. Let R be the equivalence relation defined on Z in Example 2, and write out the elements of the...
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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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