1873_solutions

An example of a set a that has a largest member a

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Unformatted text preview: w or x B, in which case x v w. Therefore no member of the set A Þ B can be greater than w and we have shown that A Þ B is bounded above. 11. a. If a man says truthfully that he sells more BMWs than anyone in the Southeast, what can you deduce about him? Solution: He is not in the Southeast. Next problem: "I sell diamonds for less than anyone in the industry." No wonder he can sell them for less. He isn’t in the industry. The diamonds are hot. b. Given that   sup A and that x  , what conclusions can you draw about the number x? Solution: The number x cannot be an upper bound of A. In other words, there must exist a member of A that is larger than x. c. Given that   inf A and that x  , what conclusions can you draw about the number x? The number x cannot be a lower bound of A. In other words, there must exist a member of A that is less than x. 12. If A and B are sets of real numbers then the sets A  B and A B are defined by a A and b B such that x  a  b AB  x and A a. Work out A  B and A B x a A and b B such that x  a b B in each of the following cases: i. A  0, 1 and B  1, 0 . We have A  B  1, 1 . Certainly, the most that a  b can be if a A and b B is 1  0  1 and the least that a  b can be if a A and b B is 0  1  1. Now if x 1, 1 then there are two possibilities: Case 1: 1 x 0. In this case, the fact that 0 A and x B and x  0  x shows that x A  B. Case 2: 0  x 1. In this case, the fact that x A and 0 B and x  x  0 shows that x A  B. ii. A  0, 1 and B  1, 2, 3 . One can use an argument similar to that used for part i to show that 0, 1  1  1, 2 or, perhaps, one could declare this fact to be obvious. Similar remarks apply to 0, 1  2 and to 0, 1  3 . Finally, A  B  0, 1  1, 2, 3  1, 2 Þ 2, 3 Þ 3, 4  1, 4 . iii. A  0, 1 and B  1, 2, 3 . We have A  B  1, 2 Þ 2, 3 Þ 3, 4 . b. Prove that if two sets A and B are bounded then so are A  B and A B. Suppose that A and B are bounded. Choose lower bounds u 1 and u 2 of A and B respectively and choose upper bounds v 1 and v 2 of A and B respectively. To show that v 1  v 2 is an upper bound of A  B, suppose that x A  B. Choose a A and b B such that x  a  b. Since a u 1 and b u 2 we have x  a  b u 1  u 2 . Thus u 1  u 2 is an upper bound of A  B and similar argument show that u 1 v 2 is an upper bound of A B and v 1  v 2 is a lower bound of 79 A  B and v 1 u 2 is a lower bound of A B. Exercises on Supremum And Infimum 1. Suppose that A is a nonempty bounded set of real numbers that has no largest member and that a that sup A  sup A a. You saw in an earlier exercise that the sets A and A A. Prove a have exactly the same upper bounds. 2. Given that A and B are sets of numbers, that A is nonempty, that B is bounded above and that A B, explain why sup A and sup B exist and why sup A sup B. Since A is nonempty and A B we know that B is nonempty. Since A B we know that any upper bound of B must be an upper bound of A. Therefore, since B is bound...
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