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\noindent
\parbox{2.25in}{\bf Math 3110-101}
\hfill {\Large \bf Homework \#3} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Sept. $14^\mathrm{th}$, 2020}

\vspace{0.2in}

\begin{enumerate}
\item A function problem
   \begin{enumerate}
   \item Let $f : \mathbb{Z} \to \mathbb{Z}$ be defined by $f(x)=2x^2-3$. 
      \begin{enumerate}
      \item Show $f$ is not 1-1.
      \item Show $f$ is not onto. 
      \item Let $A = \{-1,0,1, 2, 3\}$. Find $f(A) = \{ f(x) \;|\; x \in A \}$ \quad (the image of the set $A$ under the map $f$). 
      \item Let $A = \{-1,0,1, 2, 3\}$. Find $f^{-1}(A) = \{ x \in \mathbb{Z} \;|\; f(x) \in A \}$ \quad (the inverse image of $A$). 
      \end{enumerate}

   \vspace{0.1in}

   \item Let $g: X \to Y$. Prove that $g$ is onto if and only if \quad $g^{-1}(B) \not= \phi$ (the inverse image of $B$ is non-empty) for all non-empty subsets of $Y$: \quad $\phi \not= B \subset Y$.
   
   \vspace{0.1in}
   
   Recall that for $A \subseteq X$ and $B \subseteq Y$ \dots
   
   \[ f(A) = \{ f(x) \;|\; x \in A \} \subseteq Y \qquad \mbox{and} \qquad f^{-1}(B) = \{ x \in X \;|\; f(x) \in B \} \subseteq X \]
             
   \end{enumerate}

   \vspace{0.2in}

\item Dihedral groups: generators and relations style. Recall that \dots
\[ D_4 = \left\langle x,y \;|\; x^4=1, \  y^2=1, \mbox{ and } (xy)^2=1 \right\rangle = \left\{ 1,x,x^2,x^3,y,xy,x^2y,x^3y \right\} \]

   \begin{enumerate}
   \item Write down the Cayley table for $D_4$.
      
   \vspace{0.1in}

   \item Find the inverse of each element (i.e. $1^{-1} =???$, $x^{-1}=???$, etc.).
   
   \vspace{0.1in}

   \item Find the order of each element.
   
   \vspace{0.1in}
   
   \item Find all of the {\bf distinct} cyclic subgroups of $D_4$.
   
   \vspace{0.1in}
   
   \item What is in $Z(D_4)$ (recall that $Z(G)$ is the {\it center} of $G$)?
   
   \vspace{0.1in}
   
   \item Simplify $x^6y^{-3}x^3y^8x^{-5}yxy$.
   
   \end{enumerate}

   \vspace{0.2in}

\item The Matrix \qquad \qquad problem

   \begin{enumerate}
   \item Compute $A^{-1}B^2$ where $\displaystyle A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix} \in \mbox{GL}_2(\mathbb{Z}_9)$

   \vspace{0.1in}

   \item Find the cyclic subgroup generated by $A$. What is the order of $A$?

   \end{enumerate}

   \vspace{0.2in}

\item Let $H$ and $K$ be subgroups of a group $G$. Show that $H \cap K$ (the intersection of $H$ and $K$) is a subgroup of $G$.
\end{enumerate}

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