Show that isomophism between group is an equivalence relation. Briefly explain and show all the work.
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Show that isomophism between group is an equivalence relation. Briefly explain and show all the work.
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- 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.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 ].15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.
- 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 x2y2 is a multiple of 5.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 .Label each of the following statements as either true or false. The relation of being a homomorphic image is an equivalence relation on a collection of groups.
- 17. Find two groups and such that is a homomorphic image of but is not a homomorphic image of . (Thus the relation in Exercise does not have the symmetric property.) Exercise 15: 15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.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 3x10y is a multiple of 7.8. a. Prove that the set of all onto mappings from to is closed under composition of mappings. b. Prove that the set of all one-to-one mappings from to is closed under composition of mappings.