Let A and B be sets and f a function from A to B. Then the following assertions hold: f'is onto if and only if f is right invertible.
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- True or False Label each of the following statements as either true or false. Let and such that is onto. Then both and are onto.Prove 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.
- 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 .Label each of the following statements as either true or false. Every upper bound of a nonempty set is a least upper bound.Label each of the following statements as either true or false. Every least upper bound of a nonempty set S is an upper bound.
- 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)={x1ifxiseven2xifxisoddFor 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 onto6. For the given subsets and of Z, let and determine whether is onto and whether it is one-to-one. Justify all negative answers. a. b.