1873_solutions

And every finite set s that has n solution for each

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Unformatted text preview: exists a member m of E such that m  n. 58 From an earlier exercise we know that a subset E of Z  is infinite if and only if for every integer n there exists a member m of E such that m  n. We also know that a subset of Z  is equivalent to Z  if and only if it is infinite. 2. Given that S is an infinite set and that x S, prove that S ß S x. Since S  S x Þ x and S is infinite, the set S x must be infinite. Choose a one-one function f from Z  into S x . We now define x fn if u  f n  1 and n is a positive integer u gu  if u  f 1 if u f1 f2 f1 x Þ f Z f3 x S f2 f n1 fn fn This function g is a one-one function from S 1 fn x onto S. 3. Combine the preceding exercise and an earlier exercise to conclude that if S is a given set and x is infinite if and only if S ß S x. The desired statement follows at once. S then S 4. Suppose that S is any set and that is the family of subsets E of S for which the set S E is finite. Prove that the intersection of any two members of must belong to . Suppose that A and B belong to . Since the sets S A and S B are finite, so is the set S A Þ S B S AB and we conclude that A B . Exercises on Countability 1. Prove that if E is the set of irrational numbers then R ß E. Since the set Q of rational numbers is countable, the desired result follows at once from the theorem on removal of a countable subset from an uncountable set. 2. Suppose that A is a countable set and n is a positive integer. Suppose that B is the set of finite sequences x 1 , x 2 , , x n for which x j A for every j 1, , n . Prove that the set B is countable. Solution: For each positive integer n we define B n to be he set of finite sequences x 1 , x 2 , , x n for which x j A for every j 1, , n . We shall prove by mathematical induction that the set B n is countable for every positive integer n. Since B 1 ß A we know that B 1 is countable. Now given any positive integer n for which the set B n is countable we deduce from the fact that B n 1 ß B n A and a an earlier theorem that B n1 is countable. 3. Justify the claim that was made in the proof of that a countable union of countable sets is countable that the function h defined there is one-one. For each positive integer n, we choose a one-one function from A n into Z  and we call this function f n . Now given any Ý x  An, n1 if n is the least positive integer for which x A n , then we define 59 h x  n, f n x . We are being asked to show that the function h defined in this way is one-one. Ý Suppose that x and t belong to  n1 A n and that h t  h x . We define m to be the least positive integer for which t A m and x to be the least positive integer for which x A n . We see that m, f m t  h t  h x  n, f n x and so m  n and f m t  f m x . Since the function f m is one-one we conclude that t  x. Therefore h is one-one. 4. Suppose that n is a positive integer and that P n is the set of all polynomials that have rational coefficients and whose degrees do not exceed n. Prove that the set...
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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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