1873_solutions

Dialogue box fill in the function as f the starting

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Unformatted text preview: that f 1 f E E, suppose that x f 1 f E . Using the fact that f x f E , choose t E such that f x  f t . Since f is one-one we have x  t E. 8. Suppose that f : A B and that E B. Is it true that E  f f 1 E ? What if f is one-one? What if f is onto B? This exercise is similar to Exercise 7. No it isn’t true that E  f f 1 E and it doesn’t help to assume that f is one-one. We do have the inclusion f f 1 E E and the equation E  f f 1 E will be assured if f is onto the set B ( or even if the range of f is known to include E). 9. Suppose that f : A B and that P and Q are subsets of B. Prove the identities f 1 PÞQ  f 1 P Þf 1 Q , Solution: Given any member x of the set A, the condition x which says that either f x P or f x f 1 Q which says that either x f 1P f Q f1P f1Q, f 1 P Q f 1 P f 1 P Þ Q says that f x 1 P or x f1Q. PÞQ Q, 10. Suppose that f : A B and that P and Q are subsets of A. Which of the following statements are true? What if f is one-one? What if f is onto B? f PÞQ  f P Þf Q Hint: This statement is true. fP Q fP fQ This statement is false. Give an example. Then prove that the statement is true if f is one-one. If we define f x  x 2 for every number x then f1 1 f1 f1. In general it is clear that f P Q fP f Q . Now suppose that f is one-one and that y fP f Q . Using the fact that y f P we choose x P such that y  f x and, using the fact that y f Q we choose t Q such that y  f t . Since f x  f t and f is one-one we have x  t and so x P Q and we conclude that y f P Q . fP Q fP fQ This statement is true when f is one-one. 11. Given that f is a one-one function from A to B and that g is a one-one function from B to C, prove that the 49 function g f is one-one from A to C. Solution: We need to prove that whenever t and x are members of the set A and t x we have gft g f x. Suppose that t and x are members of the set A and that t x. Since f is one-one we have f t Therefore, since g is one-one we have g f t g f x and we have shown that g f t fx. g f x. 12. Given that f is a function from A onto B and that g is a function from B onto C, prove that the function g f is a function from A onto C. Suppose that z C. Using the fact that g is onto the set C, choose y B such that g y  z. Now we use the fact that f is onto the set B to choose x A such that f x  y. We observe that g f x  z. In this way we have shown that every member of the set C belongs to the range of the function g f. 13. Given that f : A B and that g : B C and that the function g f is one-one, prove that f must be one-one. Give an example to show that the function g does not have to be one-one. Solution: To prove that f is one-one, suppose that x 1 and x 2 are members of the set A and that x 1 x 2 . Since the function g f is one-one we know that g f x 1 g f x 2 and we see at once that f x 2 . Now we construct an example to show that the function g does not have to be one-one. We f x1 define f x  x for every x 0, 1 and we define x if x gx  0, 1 2 if 1  x 14. Given t...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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