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Commits (1)
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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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