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\noindent
\parbox{2.25in}{\bf Math 2510-410}
\hfill {\Large \bf Homework \#5} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Mar. $30^\mathrm{th}$, 2015}

\vspace{0.2in}

\begin{enumerate}
\item Let $A = \{ 15x+6y \;|\; x,y \in \mathbb{Z} \}$ and $B = \{ n \;|\; n \mbox{ is divisible by 3 }\}$. Show $A=B$.
\item Determine what the following union and intersection are. Your answer should be of form:
     $ (a,b) $, $ [a,b) $, $ (a,b] $, or $ [a,b] $ --- that is an open, half-open, or closed interval.  
     \begin{enumerate}[(a)]
     \item $ \displaystyle{ \bigcup_{n=1}^{\infty} \left( -1+\frac{1}{n}, 1+\frac{1}{n} \right) } $  
     \item $ \displaystyle{ \bigcap_{n=1}^{\infty} \left( -1+\frac{1}{n}, 1+\frac{1}{n} \right) } $  
     \end{enumerate}    
\item For each of the following functions, determine if the function is one-to-one, onto, both or neither. Prove your answers.  
     \begin{enumerate}[(a)]
     \item $ g_1 : \mathbb{R} \rightarrow \mathbb{R}_{\geq 0} $ defined by $ g_1(x)=x^2 + 1 $.  
     \item $ g_2 : \mathbb{Z} \rightarrow \mathbb{Z} $ defined by $ \displaystyle{ g_2(n) = \left\{ \begin{array}{cl} \dfrac{n}{2} & n \mbox{ is even} \\ 2n & n \mbox{ is odd} {\Large\strut} \end{array} \right. } $.  
     \item $ g_3 : \mathbb{Z} \rightarrow \mathbb{Z} $ defined by $ \displaystyle{ g_3(n) = \left\{ \begin{array}{cl} n+1 & n \mbox{ is even} \\ 2n & n \mbox{ is odd} \end{array} \right. } $.  
     \end{enumerate}
\item Let $ f:X \rightarrow Y $ be a function, let $ T_1, T_2 \subseteq Y $, and let $ S_1, S_2 \subseteq X $. Prove the following:  
     \begin{enumerate}[(a)]
     \item $ f(S_1)-f(S_2) \subseteq f(S_1-S_2) $  
     \item $ f^{-1}(T_1 \cup T_2) = f^{-1}(T_1) \cup f^{-1}(T_2) $  
     \item If $ T_1 \subseteq T_2 $, then $ f^{-1}(T_1) \subseteq f^{-1}(T_2) $  
     \end{enumerate}
\end{enumerate}

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