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Unformatted text preview: e way in the form x C y where
C is an equivalence class of ß and y G. Solution: Suppose that x is any real number. We define C to be the equivalence class of the relation
r such that x C. Since x C C we know that the ordered pair x, x C must belong to the relation r and so
x x C G. Since
x xC x xC
we have shown that there exists an equivalence class C of r and a member y of the set G such that
x x C y. Now we need to show that this decomposition of x into the form x C y is unique. Suppose that
C and D are both equivalence classes of r and that y and z are both members of the set G and that
x C y x D z.
Since
xC xD z y G
we see that the ordered pair x C , x D belongs to the relation r and we conclude that C D. Therefore
x xC y xC z
and it follows that y z.
11. Given that S is the set of all nonzero real numbers and that G 45 S and that x
G,
y
what properties must the set G have in order to make r an equivalence relation in S?
This exercise is similar to Exercise 9. The set G must be nonempty and whenever x and y belong to
G, the numbers x 1 and xy must belong to G.
r x, y S S 12. Suppose that P is the set of all positive numbers that are unequal to 1 and that Q is (as usual) the set of all
rational numbers.
a. Prove that the relation r defined by
r x, y P P log x y Q is an equivalence relation in P.
Given any number x P we have log x x 1 Q and so x r x. Therefore r is reflexive.
Given numbers x and y in P it follows from the equation
log x y 1
log y x
that log x y Q if and only if log y x Q. Therefore r is symmetric.
Given x, y and z in P, if the numbers log x y and log y z are rational then since
log y z
log y z log x y
Q
log x z
log y x
we conclude that r is transitive.
b. Suppose that E is a subset of P that contains precisely one member of each equivalence class of the
relation r. Prove that if x is any positive number unequal to 1 then x can be expressed in one and only
one way in the form x y q where y E and q is a rational number.
We deduce from Exercise 6 that whenever x P there is exactly one member y of the set E
such that log y x is rational.
13. Suppose that G is a set of nonzero real numbers and that P is the set of all positive numbers that are unequal
to 1. Prove that the following two conditions are equivalent:
a. The relation r defined by
r x, y P P log x y G is an equivalence relation in P.
b. The set G is nonempty and for all numbers a and b in the set G the number a
b belongs to G. This exercise can be completed by the methods that were used in Exercises 9 and 11.
An alternative is to use the conclusion of Exericise 11 as follows: We define
W log 2 x x P
G.
and we define the relation s in W by saying that if u and v belong to W then u s v means that u
v
Note that W is the set of nonzero real numbers. Since the condition u s v holds exactly when 2 u r 2 v
we see at once that r is an equivalence relation in P if and only if s is an equivalence relation in the
set W, the desired result follo...
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 Fall '08
 STAFF
 Math, Calculus

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