Let A # 0 be a set. Prove that A? is an equivalence relation on A.
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*NOTE: it says "Let A be a nonempty a set". It does not specify what kind of set, so we may NOT assume it is a relation. Thus A^2 means A x A, NOT composition of relations.*
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- 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.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.Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.
- In Exercises , a relation is defined on the set of all integers. In each case, prove that is an equivalence relation. Find the distinct equivalence classes of and list at least four members of each. 10. if and only if .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 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 list at least four members of each. xRy if and only if x+3y is a multiple of 4.
- 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.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.