For each of the mappings
Exhibit a left inverse whenever one exists.
For each of the following mappings
a.
c.
e.
g.
i.
k.
m.
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Elements Of Modern Algebra
- 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 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_forward5. 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.arrow_forward
- 10. Let and be mappings from to. Prove that if is invertible, then is onto and is one-to-one.arrow_forwardLet f:AA, where A is nonempty. Prove that f a has right inverse if and only if f(f1(T))=T for every subset T of A.arrow_forward4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .arrow_forward
- 3. For each of the following mappings, write out and for the given and, where.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_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_forward
- Label each of the following statements as either true or false. 3. Let , , and be mappings from into such that . Then .arrow_forwardLabel each of the following statements as either true or false. 4. Let , , and be mappings from into such that . Then .arrow_forward23. 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_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,