deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 59e2659..a4c5924 100644  --- a/untitled.tex  +++ b/untitled.tex 
...
\newtheorem{theorem}{Theorem}  這篇短文，想藉由一次小考的題目來點出大家在寫證明題時常見的錯誤；由於每個人的寫法不盡相同，這邊所提的只是一個大方向，請大家自行判斷文章所指出的，是否就是自己曾經犯過的錯誤。  \section{1. 是非題}  \begin{enumerate}  % ----------------------------------------------------------------  \section{是非題}  是非題答題方式很簡單：對的給證明，錯的給反例；但也是最難寫的題型，因為判斷對錯本身就不是一件容易的事。撇除這個部份不談，是非題比較容易犯的，嚴格來說不是錯，而是寫法上失焦。我們來看底下這個例子。  \begin{itemize}  \item If $x$ and $y$ are integers of the same parity, then $xy$ and $(x+y)^2$ are of the same parity. (Two integers are of the same parity if they are both odd or both even.)  \end{itemize}  % ----------------------------------------------------------------  \item Let $A$ and $B$ be two sets. If $A \setminus B = B \setminus A$, then $A \setminus B = \varnothing$.  % ----------------------------------------------------------------  \item Let $n \in \mathbb{Z}$. If $n^3 + n$ is even, then $n$ is even. \section{運算元混淆}  % ---------------------------------------------------------------- \begin{itemize}  \item For every positive irrational number $b$, there is an irrational number $a$ such that $0 < a < b$. Let $A$ and $B$ be two sets. If $A \setminus B = B \setminus A$, then $A \setminus B = \varnothing$.  \end{itemize}  % ----------------------------------------------------------------  \item There are infinitely many primes. \section{說明不夠詳細}  % ---------------------------------------------------------------- \begin{itemize}  \item For every positive integer $n$ irrational number $b$,  there exists is  an odd integer $m$ irrational number $a$  such that $2^{2n} + m$ is $0 <  a perfect square. < b$.  \end{itemize}  % ----------------------------------------------------------------  \item $(1 + \frac{1}{n})^n < n$ for every integer $n \geq 3$. \section{倒因為果}  % ---------------------------------------------------------------- \begin{itemize}  \item $n^3 + 1 > n^2 + n$ for every integer $n \geq 2$.  % ----------------------------------------------------------------  \item Suppose that $m, n, r \in \mathbb{N}$. If $r$ divides $mn$, then either $r$ divides $m$ or $r$ divides $n$.  % ----------------------------------------------------------------  \item Let $n \in \mathbb{N}$ be a composite number.  \begin{enumerate}  \item Suppose that $n = a b$ for some $a, b \in \mathbb{N}$. Then either $a \leq \sqrt{n}$ or $b \leq \sqrt{n}$.  \item There exists a prime factor $p$ of $n$ such that $p \leq \sqrt{n}$.  \end{enumerate}  \end{enumerate} \end{itemize}