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\title[Homework 1]{Homework 1\\
\small due: 9/14/2012}
\author{}
%\date{}                                           % Activate to display a given date or no date

\begin{document}
\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{enumerate}[1. ]
\item (D\&F 0.1.5) {\bf Determine whether the following functions $f$ are well defined:
\begin{enumerate}
	\item $f : \QQ \to \ZZ$ defined by $f(a/b) = a$.
	\item $f : \QQ \to \QQ$ defined by $f(a/b) = a^2/b^2$.
\end{enumerate}} 

%\begin{proof}[Answer.]
%
%\end{proof}

\item (D\&F 0.1.7+) {\bf Let $f: A \to X$ be a surjective map of sets. 
\begin{enumerate}
\item Prove that the relation
$$a \sim b \text{ if and only if } f(a) = f(b)$$
is an equivalence relation  (the equivalence classes are called {\sl fibers}). 
\item Why do we require that $f$ is surjective?
\item Describe the fibers when $f$ is also injective.
\end{enumerate}
}

%\begin{proof}[Answer.]
%
%\end{proof}




\item (D\&F 1.1.1-2) {\bf Of the following binary operations, determine which are (a) associative, and (b) commutative:
\begin{enumerate}[\quad i.]
	\item the operation defined on $\ZZ$ by $a\star b = a - b$;
	\item the operation defined on $\RR$ by $a\star b = a + b + ab$.
\end{enumerate}
}

%\begin{proof}[Answer.]
%
%\end{proof}


\item (D\&F 1.1.18) {\bf Determine (prove positive, or give a reason why not) which of the following sets are groups under addition:
\begin{enumerate}
	\item the set of polynomials $\ZZ[x]$;
	\item the set of rational numbers (including $0=0/1$) in lowest terms whose denominators are odd;
	\item the set of rational numbers (including $0=0/1$) in lowest terms whose denominators are even;
	\item the set of rational numbers of absolute value $< 1$;
	\item the set of rational numbers of absolute value $\geq 1$ together with 0;
	\item the set of rational numbers (in lowest terms) with denominators equal to 1 or 2;
	\item the set of rational numbers (in lowest terms) with denominators equal to 1, 2, or 3.
\end{enumerate}
}

%\begin{proof}[Answer.]
%
%\end{proof}



\item (D\&F  1.1.18) {\bf Let $x, y \in G$. Prove that $xy=yx$ if and only if $y^{-1}xy = x$ if and only if $x^{-1}y^{-1}xy=1$.
}




\item  {\bf Consider the set of functions 
$$X  = \left\{ f_{1}(x) = x, f_{2}(x) = \frac{1}{x}, f_{3}(x) = 1-x, f_{4}(x) = \frac{1}{1-x}, f_{5}(x) = \frac{-x}{1-x}, f_{6}(x) = 1 - \frac{1}{x}\right\}.$$ 
These functions are all defined on $\mathbb{R} \setminus \{0,1\}$. Prove $X$ forms a group where the operation is function composition, and write the group's multiplication table.
} 



%\begin{proof}[Answer.]
%
%\end{proof}


\item (D\&F  0.1.11-14) {\bf Showing that $(\ZZ/n\ZZ)^\times = \{ \bar{a} \in  \ZZ/n\ZZ ~|~ (a,n) = 1 \}$ is a group under multiplication: 
\begin{enumerate}
	\item Prove that if $\bar{a}, \bar{b} \in \ZZ^\times$, then $\bar{a} \cdot \bar{b} \in \ZZ^\times$. 

	\item Let $n \in \ZZ$, $n>1$, and let $a \in \ZZ$ with $1 \leq a \leq n$. Prove that if $a$ and $n$ are not relatively prime, there exists an integer $b$ with $1 \leq b < n$ such that $ab \equiv 0 \pmod{n}$ and deduce that there cannot be an integer $c$ such that $ac \equiv 1 \pmod{n}$. 

	\item Let $n$ and $a$ be as above. Prove that if $a$ and $n$ are relatively prime then there is an integer $c$ such that $ac \equiv 1 \pmod{n}$ [use the fact that the g.c.d. of two integers is a $\ZZ$-linear combination of the integers].
	
	\item Conclude from the previous two exercises that $(\ZZ/n\ZZ)^\times$ is the set of elements $\bar{a}$ of $\ZZ/n\ZZ$ with $(a,n) = 1$ and hence prove Proposition 4. Verify this directly in the case $n = 12$. 
\end{enumerate}
}

%\begin{proof}[Answer.]
%
%\end{proof}

\end{enumerate}


\end{document}