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\homework{2: More Proofs}{Due Friday, week 1}

Do \textbf{two} of the \textbf{four} problems listed here!  Prove all of your claims.


\begin{enumerate}
\item Prove or disprove the following statements.  If you disprove any statement, include an example that disproves the statement: if you prove a statement, include an example that proves the claim made.
\begin{enumerate}
\item If $x$ and $y$ are irrational, then $x+y$ is irrational.
\item If $x$ is irrational and $y$ is rational, then $x+y$ is irrational.
\item If $x$ and $y$ are rational, then $x+y$ is rational.
\item If $x$ and $y$ are irrational, then $x\cdot y$ is irrational.
\item If $x$ is irrational and $y$ is rational, then $x\cdot y$ is irrational.
\item If $x$ and $y$ are rational, then $x\cdot y$ is rational.
\item If $x$ and $y$ are rational, then $x^y$ is rational.
\end{enumerate}
\item Suppose that you have a $10 \times 10$ chessboard, and that you want to cover it with $(1 \times 4)$-sized dominoes, so that no dominoes overlap or stick off the board.  Can you do this?  Or is it impossible?  (Prove either claim.)

\item The game of \textbf{generalized $n$-tic-tac-toe} is played as follows: on a $n \times n$ grid, two players $X$ and $O$ take turns placing their respective symbols $x, o$ into cells of the grid.  No cell can be repeated.  The game ends whenever any player gets $n$ consecutive copies of their symbol on the same row /column / diagonal, or when the grid is completely filled in without any player having any such $n$ consecutive symbols.  (Normal tic-tac-toe is where $n = 3$.)  

Prove that there is no strategy in generalized tic-tac-toe where the \textbf{second player} to move is guaranteed to win.

\item  Prove that there are infinitely many prime numbers.
\end{enumerate}





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