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Unformatted text preview: hat 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 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. Suppose that f : A B and that r is the relation defined by
r x, y
A A fx fy . a. Prove that r is an equivalence relation in the set A.
Since f x f x whenever x A the relation r is reflexive. Since the condition f x f y is the
same as the condition f y f x the relation r is symmetric. A similar argument shows that r is
transitive.
b. Prove that if E is a subset of A that contains precisely one member of each equivalence class of r then the
restriction of f to E is a oneone function from E into B.
Suppose that E is a subset of A containing precisely one member of each equivalence class of
r. Given x and t different members of E we know from the fact that x and t do not lie in the same
equivalence class of r that f x
f t . Therefore the restriction of f to E is oneone.
16. Suppose that S is a given set and that is a set of functions from S to R. Prove that if for any two members f
and g of we define the condition f g to mean that f x
g x for every x S, then is a partial order of
the set .
Given any function f
we see from the fact that f x
f x for every x S that f f. Therefore
the relation in is reflexive and similar arguments can be used to show that is also symmetric
and transitive. We omit the details. 50 17. Suppose that S is a given set and that is the set of all functions from S to R. Suppose that the partial order
is defined in the set above. Given two members f and g of , prove that there exists a member u of
such that the following two conditions hold:
a. f u and g u b. Whenever a member h of satisfies f h and g h, we have u h. Suppose that f and g belong to the family . We define
ux fx if g x gx if f x g x fx It is easy to see that this function u has the desired properties.
18. 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 .
The equation f 1 f i A says that f 1 f x x for every x A and this assertion is exactly the
definition of the function f 1 . The equation f f 1 i B follows similarly.
19. 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?
If x and t are any members of A for which f x f t then it follows that
xgfx gft t
and so the function f is oneone.
Given any x A we see from the fact that x g f x that the function g is onto the set A.
b. Give...
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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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