1873_solutions

Is one one what if f is onto b f pq f p f q hint this

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Unformatted text preview: : 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 that f is a function from A onto B and that g : B both of the functions f and g have to be one-one. 5 C and that the function g f is one-one, prove that Solution: To see that f is one-one, suppose that x and t are members of the set A and that t Since g f t gfx we see at once that f t x. fx. Now to see that g is one-one, suppose that u and y are members of the set B and that u 34 y. Using the fact that the function f is onto the set B we choose members t and x of A such that u  f t and y  f x . We see at once that t x and therefore gu gft gfx gy. 15. Given any set S, the identity function i S on S is defined by i S x  x for every x S. Prove that if f is a one-one function from a set A onto a set B then f 1 f  i A and f f 1  i B . There is really nothing to prove. The fact that f 1 f x  x  iA x for every x A follows at once from the definition of the function f 1 . The equation f f 1  i B follows in a similar manner. 16. Suppose that f : A B. a. Given that there exists a function g : B A such that g f  i A , what can be said about the functions f and g? The function f must be one-one because if t and x belong to A and f t  f x then we have t  g f t  g f x  x. The function g must be onto the set A because if x is any member of A we have xgfx . b. Given that there exists a function h : B A such that f h  i B , what can be said about the functions f and h? The function f must be onto the set B and the function h must be one-one. c. Given that there exists a function g : B A such that g f  i A and that there exists a function h : B A such that f h  i B , what can be said about the functions f, g and h? From parts a and b we see that all three functions are one-one and that f is onto B and that the functions g and h are onto A. 17. As in a previous example, we define fa x  x a 1 ax whenever a 1, 1 and x 1, 1 . a. Prove that if a and b belong to 1, 1 then so does the number c  ab . 1  ab Hint: An quick way to do this exercise is to observe that c  f b a . There really isn’t much more to say here. The earlier material showed that f b is a one-one function from 0, 1 onto 0, 1 and we see at once that f b 1  1 and f b 1  1. b. Given a and b in 1, 1 and prove that f b f a  f c . Given any number x c  ab , 1  ab 0, 1 we have fb fa x  fb fa x    fb xa 1 ax x a b 1 ax 1 ax b x a x 1ab a b a b 1ab x 35  fc x   xa 1 ax 1 b b xa 1 ax x 1  ab a b 1  ab a  b x Alternative 4: A More Detailed Presentation o...
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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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