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Show that the function f : R → R,
defined as f(x) = x2 , is neither one-one nor onto.
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- 10. Let and be mappings from to. Prove that if is invertible, then is onto and is one-to-one.Let 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.4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .
- 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)={x1ifxiseven2xifxisodd27. 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 .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.
- Prove that if f is a permutation on A, then (f1)1=f.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 ontoLet g:AB and f:BC. Prove that f is onto if fg is onto.