For each of the following mappings
a.
c.
e.
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Elements Of Modern Algebra
- For each of the following mappings f:ZZ, determine whether the mapping is onto and whether it is one-to-one. Justify all negative answers. a. f(x)=2x b. f(x)=3x c. f(x)=x+3 d. f(x)=x3 e. f(x)=|x| f. f(x)=x|x| g. f(x)={xifxiseven2x1ifxisodd h. f(x)={xifxisevenx1ifxisodd i. f(x)={xifxisevenx12ifxisodd j. f(x)={x1ifxiseven2xifxisoddarrow_forwardFor 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 ontoarrow_forward3. For each of the following mappings, write out and for the given and, where.arrow_forward
- For each of the following mappings exhibit a right inverse of with respect to mapping composition whenever one exists. a. b. c. d. e. f. g. h. i. j. k. l. m. n.arrow_forward10. Let and be mappings from to. Prove that if is invertible, then is onto and is one-to-one.arrow_forward27. 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 .arrow_forward
- 2. For each of the mappings given in Exercise 1, determine whether has a left inverse. Exhibit a left inverse whenever one exists. For each of the following mappings exhibit a right inverse of with respect to mapping composition whenever one exists. a. b. c. d. e. f. g. h. i. j. k. l. m. n.arrow_forwardFor the given f:ZZ, decide whether f is onto and whether it is one-to-one. Prove that your decisions are correct. a. f(x)={ x2ifxiseven0ifxisodd b. f(x)={ 0ifxiseven2xifxisodd c. f(x)={ 2x+1ifxisevenx+12ifxisodd d. f(x)={ x2ifxisevenx32ifxisodd e. f(x)={ 3xifxiseven2xifxisodd f. f(x)={ 2x1ifxiseven2xifxisoddarrow_forwardComplete 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: .arrow_forward
- 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.arrow_forwardFor determine which of the following relations onare mappings from to, and justify your answer. b. d. f.arrow_forwardFind the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).arrow_forward
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