Prove the de nitions of addition and multiplication in Zm [x]m + [y]m = [x + y]m and [x]m[y]m = [xy]m are well de ned, that is, they are independent of the choice of representatives of the congruence classes [x]m and [y]m. In particular, let x1 = x + rm and y1 = y + sm, where r; s 2 Z. So x1 and y1 are other representatives of [x]m and [y]m. Prove that x1 + y1 x + y (mod m) and x1y1 xy (mod m):

Prove the de nitions of addition and multiplication in Zm [x]m + [y]m = [x + y]m and [x]m[y]m = [xy]m are well de ned, that is, they are independent of the choice of representatives of the congruence classes [x]m and [y]m. In particular, let x1 = x + rm and y1 = y + sm, where r; s 2 Z. So x1 and y1 are other representatives of [x]m and [y]m. Prove that x1 + y1 x + y (mod m) and x1y1 xy (mod m):

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 22E: A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which...

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