Let a relation R on R2 be given by (x,y)R(x’,y’) provided that x-x’ and y-y’ are integers. Prove that R is an equivalence relation.
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Let a relation R on R2 be given by (x,y)R(x’,y’) provided that x-x’
and y-y’ are integers. Prove that R is an equivalence relation.
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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.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.
- Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.For any relation on the nonempty set, the inverse of is the relation defined by if and only if . Prove the following statements. is symmetric if and only if . is antisymmetric if and only if is a subset of . is asymmetric if and only if .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.
- Let and be lines in a plane. Decide in each case whether or not is an equivalence relation, and justify your decisions. if and only ifand are parallel. if and only ifand are perpendicular.In Exercises , prove the statements concerning the relation on the set of all integers. 18. If and , then .Prove Theorem 1.40: If is an equivalence relation on the nonempty set , then the distinct equivalence classes of form a partition of .
- 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: .True or False Label each of the following statements as either true or false. Let be an equivalence relation on a nonempty setand let and be in. If, then.