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\title{Math 214 -- Foundations of Mathematics\\
Homework 3}
\date{Due noon Sept 21, 2012}
\author{Your name}


\begin{document} 
\maketitle

Solve the following problems. Please remember to use complete sentences
and good grammar. 

\begin{enumerate}
\item \points{4} For the following, state whether they are true or not.  Then, prove your answer.
  \begin{enumerate}
  \item $\forall x\in \R, \exists y\in \R, xy=1$;
  \item $\exists n \in \N, \exists m \in (\N - \{1\}), nm=1$.
 \end{enumerate}
 
\item \points{4} Prove that if $x$ and $y$ are positive real numbers, then $\sqrt{x+y}\not=\sqrt{x}+\sqrt{y}$. 

\item  \points{4} Prove that the product of an irrational number and a nonzero rational number is irrational.  

\item \points{4} Recall that for a given $S \subseteq \R$, the maximum element of $S$ (written $\max\{n : n \in S\}$) as the number $\alpha \in S$ such that for all $\beta \in S$, $\alpha \geq \beta$. 
  
Let $A = \{n \in \N: \sqrt{n} \not\in \Q\}$. Show that $\max\{n : n \in A\}$ does not exist.



\item \points{4} Use induction to prove that, for all $n \in \N$, $$\sum_{k=0}^n (2 k + 1) = (n+1)^2.$$

\item \points{4} Use induction to prove that for all integers $n \geq 3$, $n^3 \leq 3^n$. 
\end{enumerate}

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