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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 oneone 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 oneone? 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 oneone. 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 oneone? 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 oneone.
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 oneone 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 oneone 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 oneone.
11. Given that f is a oneone function from A to B and that g is a oneone function from B to C, prove that the 49 function g f is oneone 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 oneone we have f t
Therefore, since g is oneone 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 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 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.
 Fall '08
 STAFF
 Math, Calculus

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