1873_solutions

Could also be the set of all rational numbers between

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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.

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