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\begin{document}

\title[252 Worksheet \#1]{Math 101: Algebra I \\ Worksheet, Day \#1}
\date{Monday, 12 September 2016}

\newcommand{\shortblank}{\underline{\hspace{1in}}}
\newcommand{\longblank}{\underline{\hspace{2in}}}
\newcommand{\Longblank}{\underline{\hspace{2.5in}}}
\newcommand{\twolongblank}{\underline{\hspace{3.5in}}}
\newcommand{\prooflongblank}{\underline{\hspace{4.5in}}}
\newcommand{\fulllongblank}{\underline{\hspace{\textwidth}}}

\maketitle

We review the prerequisites for the course in set theory and beginning a first pass on group theory.  Fill in the blanks as we go along.

\section{Sets}

A \defi{set} is a ``collection of objects''.  (Our set theory is naive, and we do not go into super important foundational issues.  Please take a logic class, it is amazingly cool!)  

Basic sets:
\begin{itemize}
\item $\emptyset$, the \defi{empty set} containing no elements;
\item $\Z=\{\dots,-1,0,1,\dots\}$, the \defi{integers};
\item $\Z_{\geq 0}=\{x \in \Z : x \geq 0\}$, the nonnegative integers; similarly, positive integers, etc.;
\item $\mathbb{N}=$ \longblank, the natural numbers;
\item $\mathbb{Q}$, the rational numbers;
\item $\mathbb{R}$, the real numbers;
\item $\mathbb{C}$, the complex numbers.
\end{itemize}

A set $X$ is a \defi{subset} of a set $Y$ if $x \in X$ implies $x \in Y$, and we write $X \subseteq Y$.  (Some write $X \subset Y$.)  Two sets are equal, and we write $X=Y$, if they contain precisely the same elements, which can also be written \twolongblank.

Operations on two sets $X,Y$:
\begin{itemize}
\item $X\cup Y$, \defi{union}: we have $x \in X \cup Y$ if and only if $x \in X$ or $x \in Y$;
\item $X \cap Y$, \defi{intersection}: we have $x \in X \cap Y$ if and only if \longblank;
\item $X \smallsetminus Y$, \defi{set minus}: we have $x \in X \smallsetminus Y$ if and only if \longblank;
\item $X \sqcup Y$, \defi{disjoint union}: we write disjoint union instead of union when \\
\twolongblank.
\item $X \times Y = \{(x,y) : x \in X, y \in Y\}$, the \defi{Cartesian product}.
\end{itemize}

A \defi{relation} $R$ on a set $X$ is \twolongblank.  For example, equality is a relation on any set, defined by \Longblank.  An equivalence relation is a relation $\sim$ that is:
\begin{itemize}
\item \defi{reflexive}, \Longblank, 
\item \shortblank, \twolongblank, and
\item \shortblank, \prooflongblank.
\end{itemize}
An equivalence relation $\sim$ partitions $X$ into a disjoint union of equivalence classes, where the equivalence class of $x \in X$ is \twolongblank.  The set of equivalence classes $X/\!\sim$ is the \defi{quotient} of $X$ by $\sim$, and we have a \defi{projection} map
\begin{align*}
\pi : X \to X/\!\sim \\
x \mapsto [x]
\end{align*}

Let $n \in \Z_{>0}$.  We define an equivalence relation on $\Z$ by $x \equiv y \psmod{n}$ if $n \mid (x-y)$.  The set of equivalence classes is denoted $\Z/n\Z$.  
\newpage

\section{Functions}

A \defi{function} or \defi{map} from a set $X$ to $Y$ is denoted $f:X \to Y$: the precise definition is via its graph $\{(x,f(x)) : x \in X \} \subseteq X \times Y$.

The collection of all functions from $X$ to $Y$ is denoted $Y^X$, and this is sensible notation because \\
\fulllongblank.

Let $f:X \to Y$ be a function.  Then $X$ is the \defi{domain} and $Y$ is the \shortblank.  We write $f(X)=\img f$ for the \defi{image} of $f$.  The \defi{identity} map on $X$ is denoted $\id_X:X \to X$ and defined by \longblank.

Given another function $g:Y \to Z$, we can compose to get $g\circ f : X \to Z$ defined by $(g \circ f)(x)=g(f(x))$.  Sometimes we will have more elaborate diagrams:
\[
\xymatrix{
X \ar[r]^f \ar[dr]_h & Y \ar[d]^g \\
& Z 
} \]
We say a diagram like the above is \defi{commutative} if we start from one set and travel to any other, we get the same answer regardless of the path chosen: in the above example, this reads \longblank.  Similarly, the diagram
\[
\xymatrix{
X \ar[r]^f \ar[d]_g & Y \ar[d]^{g'} \\
X' \ar[r]^{f'} & Y' 
} \]
is commutative if and only if \longblank.

We say that $f$ \defi{factors through} a map $g:X \to Z$ if there exists a map $h:Z \to Y$ such that \twolongblank, i.e. the diagram
\vspace{1.5in}

\noindent commutes.  

The function $f$ is:
\begin{itemize}
\item \defi{injective} (or \defi{one-to-one}) if 
\twolongblank, and if so we write $X \hookrightarrow Y$; 
\item \defi{surjective} (or \defi{onto}) if
\twolongblank, and if so we write $X \twoheadrightarrow Y$; and
\item \defi{bijective} (or \defi{a one-to-one correspondence}), if $f$ is both injective and surjective, and we write $X \xrightarrow{\sim} Y$.
\end{itemize}

\begin{lem}
Define the relation $\sim$ on $X$ by $x \sim x'$ if $f(x)=f(x')$.  Then the following hold.
\begin{enumalph}
\item $\sim$ is an equivalence relation.
\item $f$ factors uniquely through the projection $\pi:X \to X/\!\sim$.  If $f$ is surjective, then the map $(X/\!\sim) \to Y$ is bijective.
\end{enumalph}
\end{lem}

In a picture:
\vspace{1.5in}


\begin{proof}
First, part (a).  {\underline{\hspace{4.9in}}} \\
\fulllongblank 

Next, part (b).  {\underline{\hspace{5.2in}}} \\
\fulllongblank \\
\fulllongblank \\
\fulllongblank \\
{\underline{\hspace{4.5in}}} \qedhere
\end{proof}

\begin{exm}
If $I$ is a set, and for each $i \in I$ we have a set $X_i$, we can form the product $X_I=\prod_{i \in I} X_i$.  The set $X_i$ has projection maps $\pi_i:X_I \to X_i$ for $i \in I$.  The product $X_I$ is uniquely determined up to bijection by the following property: for any set $Y$ and maps $f_i:Y \to X_i$, there is a unique map $f:Y \to \prod_{i \in I} X_i$ such that $\pi_i \circ f = f_i$.  In a diagram: \\
\vspace{1.5in}
\end{exm}

A \defi{left inverse} to $f$ is a function $g:Y \to X$ such that $g\circ f=\id_X$, and similarly a right inverse.  The function $f$ has a left inverse if and only if \Longblank.  In a picture:
\vspace{1in}

Similarly, $f$ has a right inverse if and only if \Longblank.

If $y \in Y$, we will write $f^{-1}(y)=\{x \in X : f(x)=y\}$ for the \defi{fiber} of $y$, and if this fiber consists of one element, we will abuse notation and also write this for the single element.

An \defi{inverse} to $f$ is a common left and right inverse.  The function $f$ has an inverse if and only if \longblank; if this inverse exists, it is unique, denoted $f^{-1}:Y \to X$ in line with the above.

The \defi{cardinality} of a set $X$ is either:
\begin{itemize}
\item \defi{finite}, if there is a bijection $X \xrightarrow{\sim} \{1,\dots,n\}$ for some $n \in \Z_{\geq 0}$, and in this case we write $\# X = n$;
\item \defi{countable}, if there is a bijection $X \xrightarrow{\sim} \Z$; or
\item \defi{uncountable}, otherwise.
\end{itemize}
If $X$ is finite, we sometimes write $\#X < \infty$ and in the latter two cases, we write $\# X=\infty$.  (This is just the beginning of a more advanced theory of cardinal numbers.)

\newpage

\section{Groups}

Let $X$ be a set.  A \defi{binary operation} on $X$ is \twolongblank.  

Let $*$ be a binary operation on $X$.  The definition is still too general, and some binary operations are better than others!  
\begin{itemize}
\item $*$ is \defi{associative} if \twolongblank.
\item $*$ \defi{has an identity} if \twolongblank.
\end{itemize}

\begin{lem}
A binary operation can have at most one identity element.
\end{lem}

\begin{proof}
\prooflongblank\shortblank
\end{proof}

\begin{defn}
A \defi{monoid} is a set $X$ equipped with an associative binary operation $*$ that has an identity.  (We will never use them, but a \defi{semigroup} is a nonempty set with an associative binary operation.)
\end{defn}

\begin{exm}
The set of positive integers $\Z_{>0}$ is a monoid under multiplication.  

The set of nonnegative integers $\Z_{\geq 0}$ is a monoid under addition.
\end{exm}

Monoids exist everywhere in mathematics, but they are still too general to study: their structure theory combines all the complications of combinatorics with algebra.  

Let $X$ be a monoid.  An element $x \in X$ is \defi{invertible} if there exists $y \in X$ such that 
\twolongblank; the element $y$ is unique if it exists because \\
\fulllongblank \\
so it is denoted $x^{-1}$ and is called the \defi{inverse} of $x$.  

\begin{defn}
A \defi{group} is a monoid in which every element is invertible.
\end{defn}

The group axioms for a group $G$ can be recovered from the requirement that $a * x=b$ has a unique solution $x \in G$ for every $a,b \in G$.

\begin{exm}
The smallest group is \Longblank, with the binary operation \longblank.  Examples of groups include:
\begin{itemize}
\item \prooflongblank\shortblank
\item \prooflongblank\shortblank
\item \prooflongblank\shortblank
\end{itemize}
\end{exm}

\begin{exm}
My favorite group is the \defi{quaternion group} of order $8$, defined by \\
\fulllongblank
\end{exm}

\begin{exm}
Let $n \in \Z_{>0}$.  The \defi{dihedral group} of order $2n$, denoted $D_{2n}$ (or sometimes $D_n$) is \\
\fulllongblank \\
\fulllongblank
\end{exm}

In a group, the \defi{(left or right) cancellation law} holds: \\
\fulllongblank

A group is:
\begin{itemize}
\item \defi{abelian} (or \defi{commutative}) if \Longblank.
\item \defi{finite} if \longblank.
\item \defi{dihedral} if \longblank.
\end{itemize}

From now on, let $G$ be a group.

\begin{lem}
If $x^2=1$ for all $x \in G$, then $G$ is abelian.
\end{lem}

\begin{proof}
\prooflongblank.
\end{proof}

The \defi{order} of an element $x \in G$ is \twolongblank, and is denoted $\shortblank$.

\begin{exm}
Important examples are matrix groups.  Let $F$ be a \defi{field}, a set with {\underline{\hspace{1.2in}}} \\ \fulllongblank.  We write $F^\times = F \smallsetminus \{0\}$.  For $n \in \Z_{\geq 1}$, let
\[ \GL_n(F)=\{A \in \mathrm{M}_n(F) : \det(A) \neq 0\} \]
be the \defi{general linear group} (of rank $n$) over $F$.  Then $\GL_n(F)$ is a group.  
\end{exm}

A \defi{homomorphism} of groups $\phi:G \to G'$ is a map such that \longblank.  Let $\phi:G \to G'$ be a group homomorphism.  Then we say $\phi$ is a(n):
\begin{itemize}
\item \defi{isomorphism} if \longblank;
\item \defi{automorphism} if \longblank;
\item \defi{endomorphism} if \longblank;
\item \defi{monomorphism} if \longblank;
\item \defi{epimorphism} if \longblank.
\end{itemize}

A \defi{subgroup} $H \leq G$ is a subset that is a group under the binary operation of $G$ (\defi{closed} under the binary operation and inverses).

\end{document}

\begin{exm}
The subgroup of $\GL_2(\C)$ generated by 
\[ A=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, B=\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} \]
is {\underline{\hspace{6.35in}}} \\
\fulllongblank
\end{exm}

