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Unformatted text preview: : B C and that the function g f is oneone, prove that f must be oneone.
Give an example to show that the function g does not have to be oneone. Solution: To prove that f is oneone, 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 oneone 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 oneone. 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 oneone. 5 C and that the function g f is oneone, prove that Solution: To see that f is oneone, 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 oneone, 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
oneone 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 oneone 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 oneone.
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 oneone 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 oneone
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
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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.
 Fall '08
 STAFF
 Math, Calculus

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