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\begin{document}
\title{Math 127 Homework}
\author{Mary Radcliffe}
\date{Due 7 February 2019}
\maketitle

Complete the following problems. Fully justify each response. You need only turn in those problems marked with a (*).

\begin{enumerate}

\item Prove the following claim. Explain what proof technique(s) you are using, and why.
\begin{quote}
Let $n$ be a positive integer. If $n$ is a perfect square, then $n$ leaves a remainder of 0 or 1 when divided by 4. 
\end{quote}

\item (*) Use the method of contrapositive to prove the Pigeonhole Principle, as stated below:
\begin{quote}
Let $m$ objects be distributed into $n$ bins. If $m>n$, then at least one bin contains at least two objects.
\end{quote}

\item (*) Prove that for all real numbers $x$, \[-5\leq |x+2|-|x-3|\leq 5.\]

\item Prove the following claim. Explain what proof technique(s) you are using, and why.
\begin{quote}
Let $a, b, c$ be positive real numbers. If $ab\geq c$, then at least one of $a, b$ is greater than or equal to $\sqrt c$. \end{quote}

\item Consider the following Theorem and Proof.
\vspace{-.2in}\begin{quote}
\begin{theorem}
For any positive integer $n$, the sum of all positive integers less than $n$ is not equal to $n$.
\end{theorem}

\begin{proof}
AFSOC that for any positive integer $n$, the sum of all positive integers is equal to $n$. But then consider $n=5$: notice that $1+2+3+4=10\neq 5$. Therefore, we have reached a contradiction, and hence for any positive integer $n$, the sum of all positive integers less than $n$ is not equal to $n$.
\end{proof}
\end{quote}

Is this result correct? Is the proof correct? What mistakes have been made?

\item Prove the following statement in three ways: directly, by contradiction, and by contrapositive. Be very explicit about the different assumptions you make in each of the three proofs. 
\begin{quote}Let $x, y$ be real numbers. If $x^2y=2x+y$, then if $y\neq 0$, then $x\neq 0$.\end{quote}

\item (*) Prove the following statement in three ways: directly, by contradiction, and by contrapositive. Be very explicit about the different assumptions you make in each of the three proofs. 
\begin{quote}Let $a, b$ be real numbers. If $a\leq x$ for every $x> b$, then $a\leq b$.\end{quote}

\item (*) Consider the following statement.
\begin{quote}Let $n$ be a positive integer, and let $y_1, y_2, \dots, y_n$ be real numbers. Let $A=\frac{y_1+y_2+\dots+y_n}{n}$ be the average of the $y_i$'s. There is at least one integer $i$ with $1\leq i\leq n$ such that $y_i\geq A$, and there is at least one integer $j$ with $1\leq j\leq n$ such that $y_j\leq A$. 
\end{quote}
Explain why you cannot prove this statement by contrapositive. Then prove it using some other technique.

\item Give an example of a proposition of the form $\forall x, \exists y, p(x, y)\wedge q(x)$. You need not be concerned with the truth value of your proposition. Write the formal negation of your statement.

\item (*) Suppose that $x$ is a member of the positive integers $\{1, 2, 3, \dots\}$ and $y$ is a member of the rationals. Let $p(x, y)$ be the statement that $xy=1$. Consider the following two statements:
\begin{enumerate}
\item $\forall x, \exists y, p(x,y)$
\item $\exists y, \forall x, p(x, y)$
\end{enumerate}
How do these statements differ? Are either of them true?

\end{enumerate}
\end{document}