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\begin{document}
\centerline{Math 112 Homework for Tuesday, Week 5}
\bigskip
\begin{enumerate}
	\item Let $F$ be a field. Prove that for any $x,y\in F$ the following equalities hold
	$$
	(-x)\cdot y=-(x\cdot y)
	$$
%	\begin{enumerate}
%		\item $(-x)\cdot y=-(x\cdot y)$
%		\item $-x-y=-(x+y)$
%		\end{enumerate}
	\item In each of the examples below, list ALL field axioms that do not
hold. Whenever possible, give a concrete example to illustrate why
the axiom does not hold.
\begin{enumerate}
\item The integers, $\Z$, with ordinary addition and multiplication.
\item The nonnegative rational numbers, $\Q_{\geq 0}$, with ordinary addition
and multiplication.
\item The set $F=\{x\}$, with just one element, and addition and multiplication defined as follows:
$$
x + x = x,\quad x\cdot x = x.
$$
\end{enumerate}
\item Let $\R$ be the set of all real numbers, and define the new addition, $\oplus$   and the new multiplication, $\otimes$ on $\R$ as follows
\begin{align*}
x\oplus y&=x+y+2\\
x\otimes y&=(x+2)(y+2)-2.
\end{align*}
Prove that $\R$ is a field with these two binary operations. {\it Hint:} determine first the additive identity, $\textbf{0}$, and multiplicative identity, $\textbf{1}$. Then you can determine the additive and multiplicative inverses, $\ominus x$ and $x^{-1}$.
\medskip

\item Let $F$ be an ordered field and let $x\in F$ satisfy $x>1$. Prove that for all positive integers $n$, we have $x^n>1$.
	\end{enumerate}

\end{document}


