M is the relation defined on Z as follows: a M b ⇐⇒ a ≡ b mod 5. (1) Prove that M is an equivalence relation. (2) Find the equivalence classe
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M is the relation defined on Z as follows:
a M b ⇐⇒ a ≡ b mod 5.
(1) Prove that M is an equivalence relation.
(2) Find the equivalence classes
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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.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. Let be an equivalence relation on a nonempty setand let and be in. If, then.
- 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.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.In Exercises , prove the statements concerning the relation on the set of all integers. 18. If and , then .
- 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.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 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.