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Unformatted text preview: that is a nested family of sets and that x Þ and y Þ , prove that there exists a member A of
such that both x and y belong to A.
Using the fact that x and y belong to Þ we choose a member A of such that x A and a member
B of such that y B. Since is nested we know that either A B or B A and in either event
we have found a member of to which both x and y belong.
4. A family of subsets of a set X is said to be field of subsets of X if
to we have X A
and A Þ B
. and for all sets A and B that belong a. Prove that if is a field of subsets of a set X and A and B are sets that belong to , then the set A
belongs to . Hint: Given any two subsets A and B of X we have
40 B BX A AÞX X B . b. Prove that if is a field of subsets of a set X and A and B are sets that belong to , then the set A
belongs to .
Given two subsets A and B of X we have
A BA
X B. B c. Prove that if is a collection of fields of subsets of a set X then the family
is also a field of subsets of
X.
Since every member of contains we know that the family Þ also contains . Given any
member A of
we know that A belongs to every member of and therefore, since every
member of is a field, X A belongs to every member of , and therefore belongs to
.
Finally, if A and B belong to
then, since both A and B belong to every member of , so does
A Þ B, which tells us that A Þ B
.
5. Suppose that is a family of subsets of a set X and that
X A is defined by
A
. a. Prove that if the intersection of any two members of belongs to then the union of any two members
of belongs to .
Suppose that the intersection of any two members of must belong to . Suppose that A and
B belong to the family . Since both X A and X B belong to we know that
XA
XB
and therefore
X
AÞB X A
XB
from which we conclude that A Þ B
.
b. Prove that if
then Þ X.
We assume that
. To prove that Þ X, suppose that x
we choose a member A of such that x A. Since
xXA
we know that x Þ .
c. Given that for every subfamily
that
.
1 Solution: Suppose that 1 1 we have Þ of 1 X. Using the fact that x , and given that 1 is a subfamily of , prove is a subfamily of . We define
X A A
1
, we have Þ 1
. Therefore
Þ1
.
1X
1 and we observe that, since 1 6. This exercise will explore some of the properties of the family that was defined in an earlier example.
Recall that is defined to be the family of all those subsets A of R for which it is possible to find a real
number a such that
A x R x a is a rational number .
a. Suppose that A is a member of the family
and only if u v is rational. and that v A. Prove that a real number u will belong to A if Solution: Choose a number a such that
A
Since v x R x a is a rational number . A we know that the number v a is rational. Now given any number u we see that
u a u v v a 41 and we deduce that the number u a will be rational if and only if the number u
other words, the condition u A holds if and only if u v is rational.
b. Prove that Þ R.
Suppose that x is any real number. We define
A t R 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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