Let R be a congruence relation modulo 7 on Z. Then the equivalence class : to which of the following: O (3] O [1] O [5] O 14] O [2]
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5
![Let R be a congruence relation modulo 7 on Z. Then the equivalence class [110] equals
to which of the following:
O [3]
O [1]
O [5]
O [4]
O [2]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7ca6c8a0-5a1f-4363-b227-340d35782c51%2F85c7031e-45f7-4bae-82f8-72dd9dee5ebd%2F2bgvsf_processed.jpeg&w=3840&q=75)
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- 5. Let be the relation “congruence modulo ” defined on as follows: is congruent to modulo if and only if is a multiple of , we write . a. Prove that “congruence modulo ” is an equivalence relation. b. List five members of each of the equivalence classes and .In this exercise set, all variables are integers. 1. List the distinct congruence classes modulo , exhibiting at least three elements in each class.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.
- a. Let R be the equivalence relation defined on Z in Example 2, and write out the elements of the equivalence class [ 3 ]. b. Let R be the equivalence relation congruence modulo 4 that is defined on Z in Example 4. For this R, list five members of equivalence class [ 7 ].In this exercise set, all variables are integers. 2. Follow the instructions in Exercise for the congruence classes modulo . 1. List the distinct congruence classes modulo , exhibiting at least three elements in each class.15. Let be a binary operation on the non empty set . Prove that if contains an identity element with respect to , the identity element is unique.
- 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.Label each of the following statements as either true or false. The distinct congruence classes for congruence modulo n form a partition of .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.
- 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.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.6. Prove that if is a permutation on , then is a permutation on .