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\title{Math 214 -- Foundations of Mathematics\\
Homework 11\\
{\large{\bf Due noon Dec 4, 2012}}
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Each problem worths 4 points unless specified otherwise. 

\begin{enumerate}
\item Let $\emptyset\not=I\subseteq \N$. For each $i\in I$, $A_i$ is denumerable.  Show that $\displaystyle \cup_{i\in I}A_i$ is denumerable. 

\item Let $A = \{(\alpha_1,\alpha_2,\alpha_3,\ldots): \alpha_i \in
  \{0,1\}, i \in \N\}$, i.e., $A$ is the infinite cartesian product of the
  set $\{0,1\}$. Show that $A$ is uncountable. 
  

\item Prove that the intervals $[0,\infty)$ and $(-1,4)$ have the same cardinality.  

\item (6 points) Determine the cardinality of the following sets (finite, denumerable, or uncountable), and justify your answers:  \\
(a) the set of all open intervals with rational midpoints. \\
(b) the set of all open intervals with rational endpoints. 
  
\item Show that $|A|<|\N|$ for every finite set $A$.   
  
\item Use Schr\"oder-Bernstein Theorem to prove that $|(2,5)|=|(0,1]|$.    


     

    

    
    
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