\documentclass[12pt]{amsart}
\theoremstyle{plain}    
\newtheorem*{definition}{Definition}
\newtheorem*{example}{Example}
\newtheorem*{exercise}{Exercise}
\newtheorem*{conjecture}{Conjecture}
\newtheorem*{theorem}{Theorem}
\newtheorem*{lemma}{Lemma}
\newtheorem*{remark}{Remark}
\begin{document} 
\title{Assignment 1}
\section{assignment 1}
\subsection{Problem 1}
Recall that if $A$ and $B$ are equivalent sets they have the same cardinality, $n(A)=n(B)$.  George Cantor extended this concept to infinite sets by defining $A$ and $B$ to be equivalent if there exists a one-to-one correspondence $f:A\rightarrow B$.  Use Cantor's definition to show that if
\[  
A=\mathbb{N}=\{1,2,3,4,\ldots\}
\]
and
\[
B=\{2,4,6,8,\ldots\}
\]
then $A\sim B$, that is, $A$ and $B$ have the same number of elements.
\subsection{Problem 2}
\par\vspace{0.3 cm}
\textit{Dirichlet's function} $f:\mathbb{R}\rightarrow\{0,1\}$ is defined by
\[
f(x)=\left\{\begin{array}{lcl}
1 &if& x\in\mathbb{Q}\;(x\mbox{ is rational})\\
0 &if& x\notin\mathbb{Q}\:(x\mbox{ is irrational})\\
\end{array}\right.
\]
\par\vspace{0.5 cm}
a) Find $f^{-1}[\{1\}]$
\par\vspace{0.5 cm}
b) Find $f[\mathbb{R}\setminus\mathbb{Q}]$
\subsection{Problem 3}
\par\vspace{0.3 cm}
Suppose $f:\mathbb{R}\setminus\{-2,2\}\rightarrow\mathbb{R}$ is defined by 
\[
f(x)=\frac{1}{x^2-4}
\]
\par\vspace{0.5 cm}
a) Find the range $f[\mathbb{R}\setminus\{-2,2\}]$
\par\vspace{0.5 cm}
b) Find the image of $(2,\infty)$, $f[(2,\infty)]$
\par\vspace{0.5 cm}
c) Find the inverse image of $\{y\;|\;1\leq y\leq 3\}$ or $f^{-1}\left[[1,3]\right]$ 
\par\vspace{0.5 cm}
d) Find the inverse image of $(-\infty,-\frac{1}{4}]$  
\subsection{Problem 4}
\par\vspace{0.3 cm}
Recall from the class notes that if $f:A\rightarrow B$ is a function and $E,G\subseteq A$,
\[
f[E\cap G]\subseteq f[E]\cap f[G]
\]
\par\vspace{0.3 cm}
Find an example of a function $f:A\rightarrow B$ and two subsets $E$ and $G$ of its domain for which
\[
f[E\cap G]\quad\mbox{is a proper subset of}\quad f[E]\cap f[G]
\]

\end{document}



