Answered: 4. For r, y E R, let z y if and only if…

4. For r, y E R, let z y if and only if (x- y) E Q. Show that defined as such is an equivale relation on R and describe the equivalence class containing 0. [

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.3: Divisibility

Problem 18E: Let R be the relation defined on the set of integers by aRb if and only if ab. Prove or disprove...

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4. For T, y e R, let z ~ y if and only if (z – y) E Q. Show that ~ defined as such is an equivale
relation on R and describe the equivalence class containing 0. [
5. Let A and B be sets with P(A) = P(B). Write a proof hy contradiction

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Let (A) be the power set of the nonempty set A, and let C denote a fixed subset of A. Define R on (A) by xRy if and only if xC=yC. Prove that R is an equivalence relation on (A).
12. (See Exercise 10 and 11.) If each is identified with in prove that . (This means that the order relation defined in Exercise 10 coincides in with the original order relation in . We say that the ordering in is an extension of the ordering in .) 11. (See Exercise 10.) According to Definition 5.29, is defined in by if and only if . Show that if and only if . 10. An ordered field is an ordered integral domain that is also a field. In the quotient field of an ordered integral domain define by . Prove that is a set of positive elements for and hence, that is an ordered field. Definition 5.29 Greater than Let be an ordered integral domain with as the set of positive elements. The relation greater than, denoted by is defined on elements and of by if and only if . The symbol is read “greater than.” Similarly, is read “less than.” We define if and only if. As direct consequences of the definition, we have if and only if and if and only if . The three properties of in definition 5.28 translate at once into the following properties of in . If and then . If and then . For each one and only one of the following statements is true: . The other basic properties of are stated in the next theorem. We prove the first two and leave the proofs of the others as exercises.
Let R be the relation defined on the set of integers by aRb if and only if ab. Prove or disprove that R is an equivalence relation.
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.