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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 oneone.
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 oneone? 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 oneone 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 oneone 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 oneone? 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 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.
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
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 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.
 Fall '08
 STAFF
 Math, Calculus

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