Define S to be the following relation on R[x]:S = {(f, g) = R[x]²: (x-3) divides (g-f) in R[x] }.You may assume that S is an equivalence relation.i. Prove that, if (f1, f2) = S and (91, 92) ES for some f₁, f2, 91, 92 € R[x], then also (f₁ + 9₁, f₂ +9₂) € S.ii. Suppose we wanted to prove that the set P of equivalence classes of S is a ring, similar to how we proved that Zm is a ring in the notes.Explain how part (i) would come up within the proof that P is a ring.

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Answered: Define S to be the following relation…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 28E

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23. Let be the equivalence relation on defined by if and only if there exists an element in such that .If , find , the equivalence class containing.
Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.
A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.
2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric or transitive. Justify your answers. a. if and only if . b. if and only if . c. if and only if for some in . d. if and only if . e. if and only if . f. if and only if . g. if and only if . h. if and only if . i. if and only if . j. if and only if . k. if and only if .
21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers. a. if and only if b. if and only if c. if and only if for some in . d. if and only if e. if and only if f. if and only if g. if and only if h. if and only if i. if and only if j. if and only if. k. if and only if.

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Define S to be the following relation on R[x]: S = {(f, g) = R[x]²: (x-3) divides (g-f) in R[x]}. You may assume that S is an equivalence relation. i. Prove that, if (f₁, f₂) € S and (9₁, 92) € S for some f₁, f2, 91, 92 € R[x], then also (f₁ + 91, f2 +9₂) € S. vivalenos simile