please send handwritten solution Q6 QUESTION 6Let S be the following relation on R:S = {(x, y) = R² :y-x is rational}.Prove that S is an equivalence relation.

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Answered: Let S be the following relation on R: S…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.1: Finite Permutation Groups

Problem 33E: Exercises 33. Prove Theorem : Let be a permutation on with . The relation defined on by if and...

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Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .
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.
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In Exercises 610, a relation R is defined on the set Z of all integers, In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and least four members of each. xRy if and only if x2+y2 is a multiple of 2.
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Let S be the following relation on R: S = {(x, y) = R² :y-x is rational}. Prove that S is an equivalence relation.