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Unformatted text preview: B is open? Solution: All we need to know is that at least one of the sets A and B is open. Suppose that A and B
are sets of real numbers, that
ABR
and that the set A is open.
To prove that
A B R,
suppose that x is any real number and that 0. Since x A we know that the set
x , x
A
is nonempty and we also know that this set is open. Therefore, since B R we know that
x , x
AB
.
We have therefore shown that every real number must belong to A B.
10. Two sets A and B are said to be separated from each other if
A BA B .
Which of the following pairs of sets are separated from each other?
a. 0, 1 and 2, 3 . Yes. b. 0, 1 and 1, 2 . Yes. c. 0, 1 and 1, 2 . No because 0, 1 d. Q and R 1, 2 1 . Q. No. 11. Prove that if A and B are closed and disjoint from one another then A and B are separated from each other.
Suppose that A and B are closed and disjoint from one another. Since A A and B B, the fact
that A B A B follows at once from the fact that A B . 101 12. Prove that if A and B are open and disjoint from one another then A and B are separated from each other.
Suppose that A and B are open and disjoint from one another. Given any number x A, we deduce
from the fact that A is a neighborhood of x and A B that x is not close to B. Therefore
A B and we see similarly that A B .
13. Suppose that S is a set of real numbers. Prove that the two sets S and R S will be separated from each other
if and only if the set S is both open and closed. What then do we know about the sets S for which S and R S
are separated from each other?
Suppose that S and R S are separated from each other. To show that S is open, suppose that
x S. Since S
R S we know that x is not close to R S. Choose 0 such that
x , x
R S
and observe that x , x
S. Thus S is open and a similar argument shows that R S is also
open. We therefore know that if the sets S and R S are separated from one another then S is both
open and closed.
Now suppose that S is both open and closed. Since the two set S and R S are closed and disjoint
from one other they are separated from one another.
14. This exercise refers to the notion of a subgroup of R that was introduced in an earlier exercise. That
exercise should be completed before you start this one.
a. Given that H and K are subgroups of R, prove that the set H K defined in the sense of
an earlier exercise is also a subgroup of R.
To prove that H K is a subgroup of R we need to show that H K is nonempty and that the
sum and difference of any members of H K must always belong to H K.
To show that H K is nonempty we use the fact that H and K are nonempty to choose x H
and y K. Since x y H K we have H K
.
Now suppose that w 1 and w 2 are any members of the set H K. Choose members x 1 and x 2 of
H and members y 1 and y 2 of K such that w 1 x 1 y 1 and w 2 x 2 y 2 . Since the numbers
x 1 x 1 and x 1 x 2 belong to H and the numbers y 1 y 2 and y 1 y 2 belong to K, and since
w1 w2 x...
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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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