Edits to group theory handout

Advanced/Group Theory/parts/00 review.tex

@@ -1,18 +1,18 @@
-\section{Review: Functions}
+\section{A Review of Functions}
 
 \definition{}
-A \textit{function} or \textit{map} $f$ from a set $A$ (the \textit{domain}, $\mathcal{D}$) to a set $B$ (the \textit{range}, $\mathcal{R}$) is a rule that assigns each element of $A$ to an element of $B$. We write this as $f: A \to B$.
+A \textit{function} or \textit{map} $f$ from a set $A$ (the \textit{domain}, $\mathcal{D}$) to a set $B$ (the \textit{range}, $\mathcal{R}$) is a rule that assigns an element of $B$ to each element of $A$. We write this as $f: A \to B$.
 
 \vspace{2mm}
 
-Consider a function $f: \mathbb{Z} \to \mathbb{Z}$. If $f(1) = 2$, we say that 2 is the \textit{image} of 1 and 1 is the \textit{preimage} of 2 under $f$.
+Consider a function $f: \mathbb{Z} \to \mathbb{Z}$. If $f(1) = 2$, we say that 2 is the \textit{image} of 1 and 1 is a \textit{preimage} of 2 under $f$.
 
 \vspace{2mm}
 
 An element in a function's domain must have exactly one image. However, an element in the range may have more than one preimage.
 
 \problem{}
-Consider the function $f: \mathbb{R} \to \mathbb{R}^+$ defined by $f(x) = x^2$
+Consider the function $f: \mathbb{R} \to \mathbb{R}^+ \cap \{0\}$ defined by $f(x) = x^2$
 \begin{itemize}
 	\item[-] What is the image of 2?
 	\item[-] What are the preimages of 9?
@@ -21,7 +21,7 @@ Consider the function $f: \mathbb{R} \to \mathbb{R}^+$ defined by $f(x) = x^2$
 \vfill
 
 \definition{}
-We say a map is \textit{one-to-one} if $a \neq b \implies f(a) \neq f(b)$. In other words, this means that each element of the range has at most one preimage.
+We say a map is \textit{one-to-one} if $a = b \implies f(a) = f(b)$ for all $a, b$ in the domain. In other words, this means that each element of the range has at most one preimage.
 
 \definition{}
 We say a map $f$ is \textit{onto} if, for every $y \in \mathcal{R}$, there exists an $x \in \mathcal{D}$ so that $f(x) = y$. In other words, this means that every $y$ in the range has a preimage in the domain.
@@ -29,7 +29,7 @@ We say a map $f$ is \textit{onto} if, for every $y \in \mathcal{R}$, there exist
 \problem{}
 Find a function that is...
 \begin{enumerate}
-	\item[-] not one-to-one, not onto
+	\item[-] neither one-to-one nor onto
 	\item[-] one-to-one and not onto
 	\item[-] not one-to-one, but onto
 	\item[-] both one-to-one and onto
@@ -53,24 +53,28 @@ Suppose $f: A \to B$ and $g: B \to C$ are both onto. Must $(g \circ f)$ be onto?
 \vfill
 \pagebreak
 
-\section{Review: Modular Arithmetic}
+
+
+
+
+\section{A Review of Modular Arithmetic}
 
 \definition{}
-$\mathbb{Z} / n$ is the set of integers mod $n$. For example, $Z/5 = \{0, 1, 2, 3, 4\}$. \\
+$\mathbb{Z}_n$ is the set of integers mod $n$. For example, $\mathbb{Z}_5 = \{0, 1, 2, 3, 4\}$. \\
 You should all be familiar with modular arithmetic.
 
 \definition{}
-The inverse of an element $a$ in $\mathbb{Z}/n$ is a $b$ so that $a \times b \equiv 1$. \\
+The inverse of an element $a$ in $\mathbb{Z}_n$ is a $b$ so that $a \times b \equiv 1$. \\
 
-Not all elements of $\mathbb{Z}/n$ have an inverse. Those that do are called \textit{units}. \\
+Not all elements of $\mathbb{Z}_n$ have an inverse. Those that do are called \textit{units}. \\
 
 \vspace{2mm}
 
-The set of all units in $\mathbb{Z}/n$ is written $(\mathbb{Z}/n)^\times$ \\
+The set of all units in $\mathbb{Z}_n$ is written $(\mathbb{Z}_n)^\times$ \\
 Read this as \say{$\mathbb{Z}$ mod $n$ cross}
 
 \problem{}
-What are the elements of $(\mathbb{Z}/5)^\times$?
+What are the elements of $(\mathbb{Z}_5)^\times$?
 
 \begin{solution}
 	$\{1, 2, 3, 4\}$
@@ -79,7 +83,7 @@ What are the elements of $(\mathbb{Z}/5)^\times$?
 \vfill
 
 \problem{}<modtables>
-Create an addition table for $\mathbb{Z}/4$ and a multiplication table for $(\mathbb{Z}/5)^\times$
+Create an addition table for $\mathbb{Z}_4$ and a multiplication table for $(\mathbb{Z}_5)^\times$
 
 \begin{center}
 \begin{tabular}{c | c c c c}
@@ -106,10 +110,10 @@ Create an addition table for $\mathbb{Z}/4$ and a multiplication table for $(\ma
 	\begin{tabular}{c | c c c c}
 		\times & 1 & 2 & 3 & 4 \\
 		\hline
-		1 & 1 & 2 & 4 & 3 \\
-		2 & 2 & 4 & 3 & 1 \\
-		3 & 4 & 3 & 1 & 2 \\
-		4 & 3 & 1 & 2 & 4 \\
+		1 & 1 & 2 & 3 & 4 \\
+		2 & 2 & 4 & 1 & 3 \\
+		3 & 3 & 1 & 4 & 2 \\
+		4 & 4 & 3 & 2 & 1 \\
 	\end{tabular}
 	\end{center}
 \end{solution}