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\begin{document}
\thispagestyle{empty}
\begin{center}
\textbf{Notation Cheat Sheet}
\end{center}
\begin{enumerate}
\item $+$ Addition
\item $-$ Subtraction
\item $\times$, $\cdot$ Multiplication
\item $\div$,$/$ Division
\item $\mathbb{N} = \{1,2,3,...\}$ Natural Numbers
\item $\mathbb{Z}$ Integers (Z\"{a}hlen)
\item $\mathbb{Q}$ Rational Numbers (Quoziente)
\item $\mathbb{R}$ Real Numbers
\item $\neg$ Not, Negation
\item $\vee$ Or
\item $\wedge$ And
\item $\equiv$ Logically Equivalent (Later: Congruences)
\item $\mid$ Divides
\item $\Rightarrow$ Implies (If... Then)
\item $\Leftrightarrow$, (iff) If and Only If
\item $\in$ In
\item $\notin$ Not In
\item $\{\}$, $\emptyset$ Empty Set
\item $\cap$ Intersection (Of Sets)
\item $\cup$ Union (Of Sets)
\item $S^{c}$, $\bar{S}$ Set Complement (With Respect to a Universe $U$)
\item $S-T$ Set Difference
\item $S \times T$ Cartesian Product
\item $\subset$ Subset
\item $\subseteq$ Subset Or Equal
\item $\subsetneq$ Proper/Strict Subset (Subset Not Equal)
\item $\supset$ Contains
\item $\supseteq$ Contains Or Equal
\item $\supsetneq$ Properly/Strictly Contains (Contains Not Equal)
\item $\forall$ For All
\item $\exists$ There Exists
\item $\exists!$ There Exists a Unique
\item $\exists^{\infty}$ There Exists Infinitely Many
\item $\#$ A contradiction
\item $\therefore$ Therefore
\item $\because$ Since  (or Because)
\item BWOC By Way of Contradiction
\item WLOG Without Loss of Generality
\item \begin{align*}
f : \mathbb{R} &\to \mathbb{N} \\
x &\mapsto \lfloor x \rfloor
\end{align*}
Here, $f$ is a function from the real numbers to the natural numbers defined by sending $x$ to the floor of $x$, that is, $f(x) = \lfloor x \rfloor$ (floor here means drop the decimal).
\item $\hookrightarrow$ Injects
\item $\twoheadrightarrow$ Surjects
\item $\displaystyle \sum_{i=1}^{n}a_{n} = a_{1} + a_{2} + ... + a_{n}$ Summation
\item $\displaystyle \sum_{i=1}^{0}a_{n} = \sum_{x \in \emptyset}x  = 0$ Empty Sum
\item $\displaystyle \prod_{i=1}^{n}a_{n} = a_{1} \cdot a_{2} \cdot ... \cdot a_{n}$ Product
\item $\displaystyle \prod_{i=1}^{0}a_{n} = \prod_{x \in \emptyset}x  = 1$ Empty Product
\item $\gcd(a,b)$ Greatest Common Divisor of $a$ and $b$
\item $\lcm(a,b)$ Least Common Multiple of $a$ and $b$
\item LDE Linear Diophantine Equations (Seeking integer solutions to an equations over the integers)
\item $a \equiv b \mod{m}$ $a$ is congruent to $b$ modulo $m$, $m \mid (a-b)$
\item $:=$ Defined To Be (Is Defined As)
\item $[a] = \{ x \in \mathbb{Z} : x \equiv a \mod{m}\}$
\item $\mathbb{Z}/m\mathbb{Z} = \mathbb{Z}_{m} = \{[0], [1], ... ,[m-1]\}$, Integers Modulo $m$
\item If $n=pq$ then $\phi(n) = \phi(pq) = (p-1)(q-1)$
\item $\mathbb{C} = \{x + yi: x,y \in \mathbb{R}\}$ Complex Numbers
\item $x + yi \in \mathbb{C}$ Standard Form of a Complex Number
\item $r(\cos(\theta) + i \sin(\theta)) \in \mathbb{C}$ Polar Form of a Complex Number
\item $\cos(\theta) + i \sin(\theta) = \text{cis}(\theta) = e^{i\theta}$ Complex Exponential
\item $\mathbb{F}$ A field (for us, fields are $\mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}_{p}$ where $p$ is a prime number)
\item $\mathbb{F}[x]$ The set (ring) of polynomials over a field $\mathbb{F}$
\item $x$ An indeterminate (sometimes called a variable)
\item $f(x) = a_{n}x^{n} + ... + a_{1}x + a_{0} \in \mathbb{F}[x]$ A polynomial
\item $a_{i}$ The coefficient of $x^{i}$
\item $a_{i}x^{i}$ A term of the polynomial $f(x)$.
\item $\mathbb{Z}[x]$ The set (ring) of polynomials over the integers (polynomials with integer coefficients)
\end{enumerate}

\end{document}
