# Proof Methods Showcase (MAT 3100 assignment template)

Author

The BFG

Last Updated

4年前

License

Creative Commons CC BY 4.0

Abstract

MAT 3100 assignment template

Author

The BFG

Last Updated

4年前

License

Creative Commons CC BY 4.0

Abstract

MAT 3100 assignment template

```
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsthm,amssymb,amsmath}
\newtheorem*{theorem}{Theorem}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\CC}{\mathbb{C}}
\title{Proof Methods Showcase}
\author{Me} % Replace Me with your name!
\date{\today}
\begin{document}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Direct Proof}
Replace this text with a few sentences describing the process for this type of proof and when you might use this method. (Reviewing Section 3.6 can help.)
\begin{theorem}
Replace this text with the theorem statement.
\end{theorem}
\begin{proof}
Write your proof here.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof by Contrapositive}
Replace this text with a few sentences describing the process for this type of proof and when you might use this method. (Reviewing Section 3.6 can help.)
\begin{theorem}
Replace this text with the theorem statement.
\end{theorem}
\begin{proof}
Write your proof here.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof by Contradiction}
Replace this text with a few sentences describing the process for this type of proof and when you might use this method. (Reviewing Section 3.6 can help.)
\begin{theorem}
Replace this text with the theorem statement.
\end{theorem}
\begin{proof}
Write your proof here.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof by Mathematical Induction}
Replace this text with a few sentences describing the process for this type of proof and when you might use this method.
\begin{theorem}
Replace this text with the theorem statement.
\end{theorem}
\begin{proof}
Write your proof here.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Some LaTeX Commands}
Here are some example sentences using LaTeX commands:\\
If $a\equiv 2\pmod{3}$, then $a^2\equiv 1\pmod 3$.\\
If $x$ and $y$ are positive real numbers, the arithmetic mean is $\dfrac{x+y}{2}$ and the geometric mean is $\sqrt{xy}$.\\
The union of two sets is $A\cup B$ and the intersection of two sets is $A\cap B$.\\
Let $(x,y)\in A\times B$.\\
Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=x^{2019}$.\\
In set-builder notation, the set of all odd integers is $\{2k+1\mid k\in\mathbb{Z}\}$.\\
Suppose that
\[1+3+5+\cdots+(2k-1) = k^2.\]
Note that if $a=2k$, then
\begin{align*}
a^2+3a+5 &= (2k)^2+3(2k)+5 \\
&= 4k^2+6k+4+1 \\
&= 2(2k^2+3k+2)+1.
\end{align*}
If $g\circ f$ is surjective, then $g$ is surjective.\\
Note that $\NN\subseteq \ZZ\subseteq \QQ\subseteq\RR\subseteq\CC$.
\end{document}
```

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