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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 oneone
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 oneone 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 oneone.
For each positive integer n, we choose a oneone 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 oneone.
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 oneone we conclude that t x. Therefore
h is oneone.
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.
 Fall '08
 STAFF
 Math, Calculus

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