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\noindent
\parbox{2.25in}{\bf Math 3110-101}
\hfill {\Large \bf Homework \#1} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Jan. $23^\mathrm{rd}$, 2015}

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\begin{enumerate}
\item Determine which of the following sets with operations are groups. If it is a group, state its identity, what the inverse of a typical element looks like, and determine if it is Abelian. If it is not a group, state which axioms hold and give counter-examples for those which fail (don't forget closure).
   \begin{enumerate}
   \item $(\mathbb{Q}_{\geq 0},+)$ non-negative rationals with addition
   \item $(\mathbb{R}_{>0},\;\cdot\;)$ positive reals with multiplication
   \item $(\mathbb{Z},-)$ integers with subtraction
   \item $(5\mathbb{Z},+)$ multiples of 5 (i.e. $0,\pm 5,\pm 10, \dots$) with addition
   \item $(\mathbb{Q}_{<0},\;\cdot\;)$ negative rationals with multiplication
   \end{enumerate}
   
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\item Let $G$ be a group. Show that $G$ is Abelian if and only if $(ab)^2=a^2b^2$ for all $a,b \in G$.

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\item Create a Cayley table for $D_3 = \{ R_0,R_{120},R_{240}, D,D',V \}$ (symmetries of an equilateral triangle). 

Is $D_3$ Abelian? 

Find the inverse of each element ($R_0^{-1} = ???$, $R_{120}^{-1} = ???$, etc.).

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