1873_solutions

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

Ask a homework question - tutors are online