Define a relation R on Z by declaring that xRy if and only if x^2 ≡ y^2 (mod 3).(a) Prove that R is an equivalence relation.(b) Find the equivalence classes [0] and [1].(c) Determine if 0 or 1 is in the equivalence class [2022]. (a) Prove that R is an equivalence relation.(b) How many distinct equivalence classes are there for R?(c) Find the smallest positive integer n such that [2022] = [n]. Define a relation R on Z by declaring that xRy if and only if x² = y? (mod 3).

(a) for any x in Z, obviously xRx, as x2 = x2 mod 3 Hence, R is reflexive. If…

Define a relation R on Z by declaring that xRy if and only if x^2 ≡ y^2 (mod 3). (a) Prove that R is an equivalence relation. (b) Find the equivalence classes [0] and [1]. (c) Determine if 0 or 1 is in the equivalence class [2022].

Define a relation R on Z by declaring that xRy if and only if x^2 ≡ y^2 (mod 3). (a) Prove that R is an equivalence relation. (b) Find the equivalence classes [0] and [1]. (c) Determine if 0 or 1 is in the equivalence class [2022].

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 20E: Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not...

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Question

Define a relation R on Z by declaring that xRy if and only if x^2 ≡ y^2 (mod 3).

(a) Prove that R is an equivalence relation.

(b) Find the equivalence classes [0] and [1].

Define a relation R on Z by declaring that xRy if and only if x² = y? (mod 3).

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