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Unformatted text preview: n that there exists a function h : B A such that f h i B , what can be said about the functions f
and h?
This is just part a again. 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?
Now the function f must be oneone and onto the set B and the functions g and h are oneone
from B onto A and, in fact g h.
20. 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 .
We saw in that earlier example that whenever 1 a 1, the function f a is a oneone function
from 1, 1 onto 1, 1 . and that f 1 1 and f 1 1. Therefore
1, 1 .
cfba b. Given a and b in 1, 1 and prove that f b f a f c .
Given any x
1, 1 we have c ab ,
1 ab 51 fb fa x xa
1 ax fb xa
1 ax 1ab
a b
1ba
a b 1 x 1 ax
1 b b xa
1 ax 1 b xa
1 ax 1 xa
1 ax b
1 ab x a b
1 ba a b x ax fc x . x Some Elementary Exercises on Set Equivalence
1. Prove that Z ß Z. Solution: The function f : Z Z defined by the equation
2n 1 if n fn 2n 0 if n 0 is a oneone function from Z onto Z .
2. Prove that 0, 1 ß 0, 1 . Solution: The function defined by the equation
1
n 1 fx if nis an integer and n 1
2 2and x if x 0 x if x 0Þ 0, 1 1
n n 1
n Zand n 2 is a oneone function from 0, 1 onto 0, 1 .
3. Prove that if a b then any two of the four intervals a, b , a, b , a, b and a, b are equivalent. Solution: At this stage we already know that
0, 1 ß 0, 1 ß 0, 1 ß 0, 1 .
If we define
f x a b a x
whenever x
0, 1 then we see that f is a oneone function from 0, 1 onto a, b and, in a similar way,
we can see that 0, 1 ß a, b and 0, 1 ß a, b and 0, 1 ß a, b .
4. Given that A and B are sets that are disjoint from each other and that A ß Z and B ß Z , prove that
A Þ B ß Z. Solution: Choose a oneone function f from A onto Z and a oneone function g from B onto Z . We
now define a function h on the set A Þ B as follows:
hx 2f x 1 if x A 2g x if x B and we observe that h is a onone function from A Þ B onto Z .
5. Given that A ß B, prove that A A ß B B.
Using the fact that A ß B we choose a oneone function f from A onto B. We now define
g x, y f x , f y 52 for every point x, y
A B.
To see that g is oneone, suppose that x, y and u, v belong to A A and that g x, y g u, v .
We have f x , f y f u , f v and so f x f u and f y f v . Since f is oneone we have x u
and y v which tells us that x, y u, v .
Now to see that g is onto the set B B, suppose that s, t
B B. Using the fact that f is onto B,
choose x and y in A such that f x s and f y t. We see that g x, y s, t . Exercises on Binary Operations
1. Prove that if we define x y x y xy for all numbers x a...
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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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