Let G= Find all the conjugate classes of (G,.). (i.e G={1,a,b,a.b}).
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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.Prove Theorem 1.40: If is an equivalence relation on the nonempty set , then the distinct equivalence classes of form a partition of .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 ].
- 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.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.Prove that if f is a permutation on A, then (f1)1=f.
- In Exercises , prove the statements concerning the relation on the set of all integers. 18. If and , then .Find all monic irreducible polynomials of degree 2 over Z3.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 .
