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......@@ -411,7 +411,7 @@ we show that the following implication holds.
\tag{$*$}
\end{equation*}
The \definend{Principle of Mathematical Induction} is that
completing both steps proves
doing both steps proves
that the statement is true for all natural
numbers greater than or equal to the initial number~$i$.
......@@ -522,7 +522,7 @@ $(2^{k+1}-1)+2^{k+1}=2\cdot 2^{k+1}-1=2^{k+2}-1$, as required.
\begin{problem}
Prove each by induction.
Suppose that $a,b\in\R$ and that $r\in\R$ with $r\neq 1$.
Suppose that $a,b,r\in\R$ and that $r\neq 1$.
\begin{exes}
\begin{exercise} \notetext{Geometric Series}
$1+r+r^2+\cdots+r^n=(r^{n+1}-1)/(r-1)$
......@@ -1420,10 +1420,9 @@ $\gcd(803,154)=\gcd(154,33)$.
Iterate:~since $154=33\cdot 4+22$, we have that
$\gcd(154,33)=\gcd(33,22)$.
Continuing gives $33=22\cdot 1+11$ and $\gcd(33,22)=\gcd(22,11)$, and
finally $22=11\cdot 2+0$ shows that
the last step is that $22=11\cdot 2+0$ shows that
$\gcd(22,11)=11$.
The zero remainder signals that the
algorithm is done, yielding $\gcd(803,154)=11$.
The zero remainder signals that we are done, and $\gcd(803,154)=11$.
Reversing this calculation is also fruitful.
Start by rewriting $33=22\cdot 1+11$ to put
......@@ -2067,7 +2066,7 @@ Together these show that $\gcd(a,b)\cdot\lcm(a,b)=ab$.)
%===================================================
\chapter{Sets}\thispagestyle{bodypage} % Can't get assignpagestyle to work
%\begin{df}
A \definend{set} is a collection that is definite, that is, well-determined,
A \definend{set} is a collection that is definite,
so that every thing either definitely is contained in the collection
or definitely is not.
An object~$x$ that belongs to a set~$A$ is an \definend{element}
......@@ -2342,7 +2341,7 @@ two sets in the universe $\universalset=\set{0,1,2,3}$.
row is the binary representation of its number.)
\begin{center} \small
\begin{tabular}{c|cc}
\multicolumn{1}{r}{\textit{number}} &$x\in A$ &$x\in B$ \\ \hline
\multicolumn{1}{r}{\textit{Number}} &$x\in A$ &$x\in B$ \\ \hline
$0$ &$0$ &$0$ \\
$1$ &$0$ &$1$ \\
$2$ &$1$ &$0$ \\
......
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