\documentclass[a4paper,10pt]{article}
\usepackage{latexsym}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{graphicx}
\usepackage{wrapfig}
\usepackage{fancybox}
\pagestyle{plain}
\begin{document}
``mcs'' --2015/5/18 --1:43 -- page 14 --\# 22
{\it 14 Chapter 1 What is a Proo}.7
1.6.1 Method \# 1: Prove Each Statement Implies the Other
The statement $P$ IFF $Q$`` is equivalent to the two statements $P$ IMPLIES $Q$'' and `` $Q$ IMPLIES $P$ So you can prove an ``iff'' by proving {\it two} implications:
1. Write, ``We prove $P$ implies $Q$ and vice-versa.'''
2. Write, ``First, we show $P$ implies $Q$ Do this by one of the methods in Section
3. Write, ``Now, we show $Q$ implies $P$ Again, do this by one of the methods in Section
1.6.2 Method \# 2: Construct a Chain of Iffs
In order to prove that $P$ is true iff $Q$ is true:
1. Write, ``We construct a chain of if-and-only-if implications.''
2. Prove $P$ is equivalent to a second statement which is equivalent to a third statement and so forth until you reach $Q.$
This method sometimes requires more ingenuity than the first, but the result can be a short, elegant proof.
Example
The {\it standard deviation} of a sequence of values $x_{1}$; $x_{2}$, . . . , $x_{n}$ is defined to be:
\begin{center}
$\sqrt{\frac{(x_{1}-\mu)^{2}+(x_{2}-\mu)^{2}++(x_{n}-\mu)^{2}}{n}}$ (1.3)
\end{center}
where $\mu$ is the average or {\it mean} of the values:
$$
\mu\ \frac{x_{1}+x_{2}+\cdots+x_{n}}{n}
$$
Theorem 1.6.1. {\it The standard deviation of a sequence of values} $x_{1}$, . . . , $x_{n}$ {\it is zero iff all the values are equal to the mean}.
For example, the standard deviation of test scores is zero if and only if everyone scored exactly the class average.
{\it Proo}. We construct a chain of ``iff'' implications, starting with the statement that the standard deviation $()$ is zero:
\begin{center}
$\sqrt{\frac{(x_{1}-\mu)^{2}+(x_{2}-\mu)^{2}++(x_{n}-\mu)^{2}}{n}}=0$. (1.4)
\end{center}\end{document}