Let R be a relation on the set of all integers such that aRb if and only if a + b is even. Is R symmetric? If yes, prove it; if no, give a counterexample.
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- Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.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.
- 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.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.In Exercises , prove the statements concerning the relation on the set of all integers. 17. If and , then .
- 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.True or False Label each of the following statements as either true or false. If is an equivalence relation on a nonempty set, then the distinct equivalence classes of form a partition of.Label each of the following statements as either true or false. If R is an equivalence relation on a nonempty set A, then any two equivalence classes of R contain the same number of element.