1873_solutions

Any such n we have f n x fx x1 x2 x3 and so the

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Unformatted text preview: i . Do you think that the axiom of choice is needed to produce a choice function relative to this association? Solution: Yes, the axiom of choice is needed here. 4. Suppose that to each member i of a given set I there is associated a nonempty well ordered set A i . Do you think that the axiom of choice is needed to produce a choice function relative to this association? Solution: No, the axiom of choice is not needed. We can give a specific definition of a choice function f by defining f i to be the least member of the set A i for each i I. 5. Given a finite set A, it is easy to give A a well order. Thus, if to each member i of a given set I there is associated a nonempty finite set A i , then we can assign a well order to A i for each i and then define f i to be the least member of A i for each i. In view of this fact, do you want to change your mind about the answer you gave for the above exercise? Solution: No you do not want to change your mind. The fact that each set A i can be given a well order is not the same as having each set A i provided with a specific well order. In order to make use of the fact that each set A i can be given a well order it is necessary to choose a well order of A i for each i and this process requires the axiom of choice. 6. Suppose that I is a given set and that for each i I we have A i  0, 1 . Do you think that the axiom of choice is needed to produce a choice function relative to this association? 7. Suppose that I is a given set and that for each i I we have A i  R. Do you think that the axiom of choice is needed to produce a choice function relative to this association? Solution: No the axiom of choice is not needed. We can obtain a choice function f very simply by defining f i  0 for each i I. 8. Given that A and B are nonempty sets, do we need the axiom of choice to guarantee that the set A nonempty? Solution: No, the axiom of choice is not needed. To show that A then choose y B. The ordered pair x, y must belong to A B is nonempty, choose x B is A and B. 9. Using the axiom of choice, prove that there exists a subset S of R such that the following two conditions hold: a. Whenever x and y belong to S and x y, the number x y is irrational. b. For every real number x there exists a member y of S such that the number x 66 y is rational. Solution: We define a relation ß in R by saying that if x and y are any real numbers then the condition x ß y means that the number x y is rational. The relation ß is an equivalence relation in R. We express the family of equivalence classes of this relation as E i i I . Using the axiom of E i for choice we choose a choice function f relative to the association of i to E i . In other words, f i each i. We define S  f i i I . Since every number must lie in precisely one equivalence class of the relation ß we know that if x R then there is precisely one member i I for which x f i is rational. Furthermore, if i j then the fact that f i and f j lie in different equivalence cl...
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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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