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Unformatted text preview: and observe that x A v is rational. In x is rational . c. Prove that if A and B are any two different members of then A B .
Suppose that A and B are members of and that A B. We may assume, without loss of
generality that A B
. Choose x A B. Given any member Y of the set B we know from
part a of this exercise and from the fact that x B that the number x y can’t be rational; and
therefore we know that y x can’t be rational. Thus no member y of the set B can belong to A
and we have shown that A B .
d. Suppose that we have selected exactly one number in each member A of and have then collected these
numbers together to form a set E. Suppose that for each rational number r we have defined
Er r x x E .
Prove that
Þ Er r Q R
and that whenever r and s are different rational numbers then E r E s .
Suppose that u is any real number. Using the fact that Þ R we choose a member A of
such that u A. We know that the set E has exactly one member that belongs the set A and we
call this member x. Since
u x u x
and since u x is rational we know that u E u x .
Finally suppose that r and s are rational numbers and that E r E s
. We shall show that
r s. Choose a number u E r E s . Choose x and y in the set E such that
u r x s y.
Since x y s r Q we know that x and y must belong to the same member of the family
and therefore, from the way in which the set E was specified we conclude that x y. The
equation r x s y now guarantees that r s, as promised. Exercises on Relations
1. Under what conditions do we have x, y y, x ?
The condition x, y y, x implies that x y and is obviously true in this case.
2. For each of the relations defined earlier, find its domain and the range.
a. We write P for the set of all people and define
P
r x, y P x is a brother of y . The domain of r is the set of all those males who have at least one sibling. The range of r is the
set of all those people who have at least one male sibling.
b. We write P for the set of all people and define
PP
r x, y x is a blood relation of y . c. We suppose that J is a bag (set) of jelly beans and define
J J x and y have the same same color .
r x, y
Both the domain and range of r are the entire set J.
d. Writing R for the set of all real numbers, we define 42 r x, y
RR
Both the domain and range of r are the entire set R.
e. Writing Z for the set of all integers, we define
ZZ
r x, y x y. Z such that y nx . n Both the domain and range of r are the entire set Z.
f. We define
r x, y R R x y is rational . Both the domain and range of r are the entire set R.
g. We define
r 0, 1 , 0, 2 , 4, 1 , 3, 2 .
Note that r is a subset of 0, 4, 3
1, 2 .
The domain of r is 0, 4, 3 and the range of r is 1, 2 .
h. We define
r x, y R R x2 y2 1 . where R is the set of all real numbers.
Given any numbers x and y, the equation x 2 y 2 1 requires that 1
Both the domain and range of r are equal to 1, 1 . x 1 and 1 y 1. i. Suppose that S is any set and define
r x, y...
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 Fall '08
 STAFF
 Math, Calculus

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