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%%% exercise set 2
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Math 445,  Fall 2009 \hfill  Exercise Set \#2 \hfill Turn in: September 11 
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 {\bf 1.}   [\#1, page 26] ~   Let $A$ be a countable set and suppose there exists a function $f \colon A \to B$ which is surjective. 
 Prove that $B$ is also countable. (Recall that a set is \emph{countable} if there is a bijection with the set of  natural numbers $\mN$.)
  
  
  
  
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 {\bf 2.} [\#4, page 26]   ~ Show that the set $\mN$ of natural numbers can be represented as a union $\mN = \cup A_i$ of an infinite number of disjoint \emph{infinite} sets. 
 
 
 
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 {\bf 3.} [\#10, page 27]   ~ Let $A$ be an infinite set, $B \subset A$ a finite subset, and $C = A - B$ the complement of $B$ in $A$. 
 Prove there exists a one-to-one correspondence between $A$ and $C$.
 
 
 
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 {\bf 4.} [\#11*, page 27]   ~  Let $A$ be an uncountable set, $B \subset A$ a countable subset, and $C = A - B$ the complement of $B$ in $A$. 
 Prove there exists a one-to-one correspondence between $A$ and $C$.

 


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 {\bf 5.} [\#1, page 31]     ~ Prove that the set of positive real numbers has cardinal $c$. (Recall that $c$ is the cardinal of the real number line $\mR$.)
 


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 {\bf 6.} [\#5, page 31]    ~ What is the cardinal number of the set of irrational numbers? 
 Of the set of transcendental real numbers? 
 (Recall that a real number is \emph{transcendental} if it is not algebraic. A real number is \emph{algebraic} if it is the solution of a non-trivial polynomial equation with integer coefficients.)
 
 
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 {\bf 7.}   [\#2, page 39]    ~ Let $L$ be a lattice in which every chain has an upper bound. Prove that $L$ has a unique maximal element; that is, a top element. 
 (You can assume Zorn's Lemma.)
  
 

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