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].

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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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