1873_solutions

Ask scientific notebook to solve the equation f g x g

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Unformatted text preview: erefore impossible. If y 3 then the equation x 3 y  2y says that 2y . x 3y We conclude that the range of f is R 3 and that for every number y R 3 there is a unique number x for which y  f x . Therefore the function f is one-one. b. Point at the equation y  3x 2 x1 and ask Scientific Notebook to solve for x. How many values of x are given? Is this result consistent with the answer you gave in part a of the question? 7. Suppose that f : A B and that E A. Is it true that E  f 1 f E ? What if f is one-one? What if f is onto B? We certainly have E f 1 f E but the inclusion can be strict. For example, if we define f x  x 2 for every number x then f 1 f 0, 1 0, 1 . Now suppose that f is a one-one function from a set A to a set B and that E A. We shall prove E and, for this purpose, we suppose that x f 1 f E . We know that f x f E and, that f 1 f E using this fact, we choose a member t of the set E such that f x  f t . Since f is one-one we have x  t and so x E which is what we needed to show. 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? No it isn’t true. There is no reason to suppose that every member of E has to be in the range of f. For example, if we define f x  x 2 for every number x and E  1, 1 Then E f f 1 E . In the 33 event that f is onto the set B the equation E  f f 9. Suppose that f : A 1 will hold. E 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. 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 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 the function g is onto the set C, choose a member y of the set B such that z  g y . Now, using the fact that the function f is onto the set B we choose a member x of the set A such that y  f x . We have found a member x of A such that z  f g x . Therefore C is the range of the function f g. 13. Given that f : A B and that g...
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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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