(a) Prove that there exists a function f: Z → Z so that f is one-to-one but f is not onto. (b) Prove that there exists a function g : R → R so that g is onto but g is not one-to-one.
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- 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.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)={x1ifxiseven2xifxisoddProve that if f is a permutation on A, then (f1)1=f.
- Let a and b be constant integers with a0, and let the mapping f:ZZ be defined by f(x)=ax+b. Prove that f is one-to-one. Prove that f is onto if and only if a=1 or a=1.Let be a field. Prove that if is a zero of then is a zero ofGive an example of mappings and such that one of or is not onto but is onto.
- 27. 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 .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.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: .
- 6. a. Give an example of mappings and , different from those in Example , where is one-to-one, is onto, and is not one-to-one. b. Give an example of mappings and , different from Example , where is one-to-one, is onto, and is not onto.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.Let f: X→Y and g: Y→Z be functions. Prove that if f is an onto function and g∘f is a one-to-one function, then g is also a one-to-one function.