QUESTION 2 Let S be the following relation on C: S = {(x, y) = C²:y-x is real}. Prove that S is an equivalence relation.
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- 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.Exercises 33. Prove Theorem : Let be a permutation on with . The relation defined on by if and only if for some is an equivalence relation on .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: .
- 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 .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.
- Label each of the following statements as either true or false. Every mapping on a nonempty set A is a relation.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.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.