If h:(R,+) → (R*,.) is a mapping defined by h(x)=10*, VXER, then (R,+) = (R*,.).
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- For each of the following parts, give an example of a mapping from E to E that satisfies the given conditions. a. one-to-one and onto b. one-to-one and not onto c. onto and not one-to-one d. not one-to-one and not onto27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].
- Prove that if a subring R of an integral domain D contains the unity element of D, then R is an integral domain. [Type here][Type here]4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .If e is the unity in an integral domain D, prove that (e)a=a for all aD. [Type here][Type here]
- Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .5. For each of the following mappings, determine whether the mapping is onto and whether it is one-to-one. Justify all negative answers. (Compare these results with the corresponding parts of Exercise 4.) a. b. c. d. e. f.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.