Created the Abstract Algebra theorems and definitions cheat sheet

part-1/chapters/chapter-0/chapter-0.tex

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+\chapter{Preliminaries}
+\subimport{./}{properties-of-integers.tex}
+\subimport{./}{modular-arithmetic.tex}
+\subimport{./}{complex-numbers.tex}
+\subimport{./}{mathematical-induction.tex}
+\subimport{./}{equivalence-relations.tex}
+\subimport{./}{functions-mappings.tex}

part-1/chapters/chapter-0/complex-numbers.tex

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+\section{Complex Numbers}
+\begin{theorem}[Properties of Complex Numbers]
+	\hfill
+	\begin{enumerate}
+		\item Closure under addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$
+		\item Closure under multiplication: $(a + bi)(c + di) = (ac) + (ad)i + (bc)i + (bd)i^2 = (ac - bd) + (ad + bc)i$
+		\item Closure under division ($c + di \neq 0$): $\displaystyle\frac{(a + bi)}{(c + di)} = \frac{(a + bi)}{(c + di)}\frac{(c - di)}{(c - di)}=\frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} = \frac{(ac + bd)}{c^2 + d^2} + \frac{(bc - ad)}{c^2 + d^2}i$
+		\item Complex conjugation: $(a + bi)(a - bi) = a^2 + b^2$
+		\item Inverses: For every nonzero complex number $a + bi$ there is a complex number $c + di$ such that $(a + bi)(c + di)=1$. (That is, $(a + bi)^{-1}$ exists in $\C$.)
+		\item Powers: For every complex number $a + bi = r(\cos\theta + i \sin \theta)$ and every positive integer $n$, we have $(a + bi)^n = [r(\cos \theta + i \sin \theta)]^n = r^n(\cos n \theta + i \sin n \theta)$.
+		\item Radicals: For every complex number $a + bi = r(\cos \theta + i \sin \theta)$ and every positive integer $n$, we have $\displaystyle(a + bi)^{\frac{1}{n}} = [r(\cos \theta + i \sin \theta)]^{\frac{1}{n}} = r^{\frac{1}{n}}(\cos\frac{\theta}{n} + i \sin \frac{\theta}{n})$.
+	\end{enumerate}
+\end{theorem}
+

part-1/chapters/chapter-0/equivalence-relations.tex

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+\section{Equivalence Relations}
+
+\begin{definition}[Equivalence Relation]
+	An \textit{equivalence relation} on a set $S$ is a set $R$ of ordered pairs of elements of $S$ such that
+
+	\begin{enumerate}
+		\item $(a, a) \in R$ for all $a \in S$  (reflexive property).
+		\item $(a, b) \in R$ implies $(b, a) \in R$  (symmetric property).
+		\item $(a, b) \in R$ and $(b, c) \in R$ imply $(a, c) \in R$  (transitive property).
+	\end{enumerate}
+\end{definition}
+
+\begin{definition}[Partition]
+	A \textit{partition} of a set $S$ is a collection of nonempty disjoint subsets of $S$ whose union is $S$.
+\end{definition}
+
+\begin{theorem}[Equivalence Classes Partition]
+	The equivalence classes of an equivalence relation on a set $S$ constitute a partition of $S$. Conversely, for any partition $P$ of $S$, there is an equivalence relation on $S$ whose equivalence classes are the elements of $P$.
+\end{theorem}

part-1/chapters/chapter-0/functions-mappings.tex

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+\section{Functions (Mappings)}
+
+\begin{definition}[Function (Mapping)]
+	A \textit{function} (or \textit{mapping}) $\phi$ from a set $A$ to a set $B$ is a rule that assigns to each element $a$ of $A$ exactly one element $b$ of $B$. The set $A$ is called the \textit{domain of $\phi$}, and $B$ is called the \textit{range of $\phi$}. If $\phi$ assigns $b$ to $a$, then $b$ is called the \textit{image of $a$ under $\phi$}. The subset of $B$ comprising all the images of elements of $A$ is called the \textit{image of $A$ under $\phi$}.
+\end{definition}
+
+\begin{definition}[Composition of Functions]
+	Let $\phi: A \to B$ and $\psi: B \to C$. The \textit{composition $\psi\phi$} is the mapping from $A$ to $C$ defined by $(\psi\phi)(a) = \psi(\phi(a))$ for all $a$ in $A$.
+\end{definition}
+
+\begin{definition}[One-to-One Function]
+	A function $\phi$ from a set $A$ is called \textit{one-to-one} if for every $a_1, a_2 \in A,\ \phi(a_1) = \phi(a_2)$ implies $a_1 = a_2$.
+\end{definition}
+
+\begin{definition}[Functions from A onto B]
+	A function $\phi$ from a set $A$ to a set $B$ is said to be \textit{onto $B$} if each element of $B$ is the image of at least one element of $A$. In symbols, $\phi: A \to B$ is onto if for each $b$ in $B$ there is at least one $a$ in $A$ such that $\phi(a) = b$.
+\end{definition}
+
+\begin{theorem}[Properties of Functions]
+	Given functions $\alpha: A \to B$, $\beta: B \to C$, and $\gamma: C \to D$, then
+	\begin{enumerate}
+		\item $\gamma(\beta\alpha) = (\gamma\beta)\alpha$ (associativity).
+		\item If $\alpha$ and $\beta$ are one-to-one, then $\beta\alpha$ is one-to-one.
+		\item If $\alpha$ and $\beta$ are onto, then $\beta\alpha$ is onto.
+		\item If $\alpha$ is one-to-one and onto, then there is a function $\alpha^{-1}$ from $B$ onto $A$ such that $(\alpha^{-1}\alpha)(a) = a$ for all $a$ in $A$ and $(\alpha\alpha^{-1})(b) = b$ for all $b$ in $B$.
+	\end{enumerate}
+\end{theorem}

part-1/chapters/chapter-0/mathematical-induction.tex

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+\section{Mathematical Induction}
+
+\begin{theorem}[First Principle of Mathematical Induction]
+	Let $S$ be a set of integers containing $a$. Suppose $S$ has the property that whenever some integer $n \geq a$ belongs to $S$, then the integer $n + 1$ also belongs to $S$. Then, $S$ contains every integer greater than or equal to $a$.
+\end{theorem}
+
+\begin{theorem}[Second Principle of Mathematical Induction]
+	Let $S$ be a set of integers containing $a$. Suppose $S$ has the property that $n$ belongs to $S$ whenever every integer less than $n$ and greater than or equal to $a$ belongs to $S$. Then, $S$ contains every integer greater than or equal to $a$.
+\end{theorem}

part-1/chapters/chapter-0/modular-arithmetic.tex

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+\section{Modular Arithmetic}

part-1/chapters/chapter-0/properties-of-integers.tex

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+\section{Properties of Integers}
+\begin{aside}[Well Ordering Principle]
+	Every nonempty set of positive integers contains a smallest number.
+\end{aside}
+
+\begin{theorem}[Division Algorithm]
+	Let $a$ and $b$ be integers with $b > 0$. then there exist unique integers $q$ and $r$ with the property that $a = bq + r$, where $0 \leq r < b$.
+\end{theorem}
+
+\begin{definition}[Greatest Common Divisor, Relatively Prime Integers]
+	The \textit{greatest common divisor} of two nonzero integers $a$ and $b$ is the largest of all common divisors of $a$ and $b$. We denote this integer by $\gcd(a, b)$. When $\gcd(a, b) = 1$, we say that $a$ and $b$ are \textit{relatively prime}.
+\end{definition}
+
+\begin{theorem}[GCD Is a Linear Combination]
+	for any nonzero integers $a$ and $b$, there exist integers $s$ and $t$ such that $\gcd(a, b)=as+bt$. Moreover, $\gcd(a,b)$ is the smallest positive integer of the form $as + bt$.
+\end{theorem}
+
+\begin{corollary}
+	If $a$ and $b$ are relatively prime, then there exist integers $s$ and $t$ such that $as + bt = 1$.
+\end{corollary}
+
+\begin{lemma}[Euclid's Lemma \text{\normalfont $p\ \vert\ ab$ implies $p\ \vert\ a$ or $p\ \vert\ b$}]
+	If $p$ is a prime that divides $ab$, then $p$ divides $a$ or $p$ divides $b$.
+\end{lemma}
+
+\begin{theorem}[Fundamental Theorem of Arithmetic]
+	Every integer greater than 1 is a prime or a product of primes. this product is unique, except for the order in which the factors appear. That is, if $n = p_1p_2\dots p_r$ and $n=q_1q_2\dots q_s$, where the $p$'s and $q$'s are primes, then $r = s$ and, after renumbering the $q$'s, we have $p_i = q_i$ for all $i$.
+\end{theorem}
+
+\begin{definition}[Least Common Multiple]
+	The \textit{least common multiple} of two nonzero integers $a$ and $b$ is the smallest positive integer that is a multiple of both $a$ and $b$. We will denote this integer by $\lcm(a, b)$.
+\end{definition}
+

part-1/part-1.tex

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+\part{Integers and Equivalence Relations}
+\subimport{chapters/chapter-0}{chapter-0.tex}
+