Introduction to Abstract Math Weekly Homework #1
Author
Joseph Stockdill
Last Updated
3 years ago
License
Creative Commons CC BY 4.0
Abstract
weekly homework
weekly homework
% --------------------------------------------------------------
% This is all preamble stuff that you don't have to worry about.
% Head down to where it says "Start here"
% --------------------------------------------------------------
\documentclass[12pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amsthm,amssymb}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newenvironment{theorem}[2][Theorem]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{lemma}[2][Lemma]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{exercise}[2][Exercise]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{problem}[2][Problem]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{question}[2][Question]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{corollary}[2][Corollary]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{solution}{\begin{proof}[Solution]}{\end{proof}}
\begin{document}
% --------------------------------------------------------------
% Start here
% --------------------------------------------------------------
\title{Weekly Homework 1}%replace X with the appropriate number
\author{Joey Stockdill\\ %replace with your name
Introduction to Abstract Math} %if necessary, replace with your course title
\maketitle
\begin{theorem}{2.15} %You can use theorem, exercise, problem, or question here. Modify x.yz to be whatever number you are proving
An odd integer times an odd integer results in an odd integer.
\end{theorem}
\begin{proof} %You can also use solution in place of proof.
Assume m and n are both odd integers.\\
Let m = 2k+1, and n = 2j+1\\
So mn = 4kj+2k+2j+1\\
Which factors into 2(2kj+k+j)+1.\\
%Note 1: The * tells LaTeX not to number the lines. If you remove the *, be sure to remove it below, too.
%Note 2: Inside the align environment, you do not want to use $-signs. The reason for this is that this is already a math environment. This is why we have to include \text{} around any text inside the align environment.
By definition 2.9 (2kj+k+j) is an integer.\\
so by definition 2.10 mn is an odd integer.\\\\
\end{proof}
\begin{theorem}{2.12} %You can use theorem, exercise, problem, or question here. Modify x.yz to be whatever number you are proving
The product of an odd integer and an even integer is odd.\\\\
\end{theorem}
\begin{proof} %You can also use solution in place of proof.
The product of 2 times 3 is 6, which is an even number.
\end{proof}
% --------------------------------------------------------------
% You don't have to mess with anything below this line.
% --------------------------------------------------------------
\end{document}