1873_solutions

# X y z r then we can see that r 3 r z r and so it

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Unformatted text preview: n: Suppose that f is any function from S to the set S Z . Given any member x of S we know that f x is a function from Z  to S which means that f x is a sequence of members of S. For each positive integer n and each x S we define f n x to be the nth member of the sequence f x . In other words, if x S then f x is the sequence f1 x , f2 x , f3 x ,  .  We need to show that the range of f must be a proper subset of S Z and for this purpose we shall find a sequence x 1 , x 2 , x 3 ,  of members of S such that x 1 , x 2 , x 3 ,  is not in the range of f. For each positive integer n we use the fact that S fn Sn to choose a member x n of S such that xn S fn Sn Now given any member x of the set S it follows from the fact that S  Sn n Z S n . For any such n we have f n x fx x1, x2, x3,  and so the sequence x 1 , x 2 , x 3 ,  has the desired properties. that for some n we have x x n and therefore  b. Prove that if S is the gigantic set defined earlier then the sets S and S Z are not equivalent. 18. Suppose that to each member i of a given set I there is associated a set S i that is strictly subequivalent to a given set U. Prove that  i I S i is strictly subequivalent to the set U I . Solution: Suppose that f is a function from  i I S i to U I and for each i fi : i I Si I we define a function U by the equation fi x  f x i for every x  i I S i . For each i we use the fact that f i S i U to choose a member that we shall call g i of the set U such that g i U f S i . In this way we have defined a member g of U I . To see that g does not lie in the range of f we observe that if x  i I S i then for some i we have x S i and for any such i we have fx i gi. Exercises on The Axiom of Choice 65 1. One of the assertions of an earlier theorem was that if A and B are given sets and if there exists a function g from B onto A then there must exist a one-one function from A into B. Rewrite the proof of this part of the theorem and show how and where the axiom of choice is used. Solution: We assume that g is a function from B onto A. For every member x of the set A we define Ex  y B g y  x . Since the function g is onto A we know that all of the sets E x are nonempty. The axiom of choice therefore guarantees the existence of a function f defined on A such that f x E x for every x A. We observe that for every member x of the set A we have g f x  x. To see that the function f is one-one, suppose that x 1 and x 2 belong to A and that f x 1  f x 2 . We see that x1  g f x1  g f x2  x2. 2. Suppose that I is a given set and that to each member i I there is associated a nonempty set A i of natural numbers. Explain why the axiom of choice does not have to be used to produce a choice function relative to this association. Solution: We can provide a specific definition of a choice function in this example by defining f i be the least member of the set A i for each i to I. 3. Suppose that to each member i of a given set I there is associated a nonempty finite set A...
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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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