X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=examples.tex;h=e11815ef7856b47f0dd5cb0e775c3272caeec427;hb=71dc84ac39e0dc87daf64fe61d7523e33711dc34;hp=c1f3df1eb64e416fb00c52f809cdcfc562138324;hpb=71b85fe2012cdf733f6e72137038d7d9960ddf08;p=mjotex.git

diff --git a/examples.tex b/examples.tex
index c1f3df1..e11815e 100644
--- a/examples.tex
+++ b/examples.tex
@@ -29,7 +29,13 @@
     If $R$ is a \index{commutative ring}, then $\polyring{R}{X,Y,Z}$
     is a multivariate polynomial ring with indeterminates $X$, $Y$,
     and $Z$, and coefficients in $R$. If $R$ is a moreover an integral
-    domain, then its fraction field is $\Frac{R}$.
+    domain, then its fraction field is $\Frac{R}$. If $x,y,z \in R$,
+    then $\ideal{\set{x,y,z}}$ is the ideal generated by
+    $\set{x,y,z}$, which is defined to be the smallest ideal in $R$
+    containing that set. Likewise, if we are in an algebra
+    $\mathcal{A}$ and if $x,y,z \in \mathcal{A}$, then
+    $\alg{\set{x,y,z}}$ is the smallest subalgebra of $\mathcal{A}$
+    containing the set $\set{x,y,z}$.
   \end{section}
 
   \begin{section}{Algorithm}
@@ -53,8 +59,9 @@
   \end{section}
 
   \begin{section}{Arrow}
-    The identity operator on $V$ is $\identity{V}$. The composition of
-    $f$ and $g$ is $\compose{f}{g}$. The inverse of $f$ is
+    The constant function that always returns $a$ is $\const{a}$. The
+    identity operator on $V$ is $\identity{V}$. The composition of $f$
+    and $g$ is $\compose{f}{g}$. The inverse of $f$ is
     $\inverse{f}$. If $f$ is a function and $A$ is a subset of its
     domain, then the preimage under $f$ of $A$ is $\preimage{f}{A}$.
   \end{section}
@@ -67,9 +74,31 @@
   \begin{section}{Common}
     The function $f$ applied to $x$ is $f\of{x}$. We can group terms
     like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a
-    set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. Here's a pair
-    of things $\pair{1}{2}$ or a triple of them $\triple{1}{2}{3}$,
-    and the factorial of the number $10$ is $\factorial{10}$.
+    set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The tuples go
+    up to seven, for now:
+    %
+    \begin{itemize}
+      \begin{item}
+        Pair: $\pair{1}{2}$,
+      \end{item}
+      \begin{item}
+        Triple: $\triple{1}{2}{3}$,
+      \end{item}
+      \begin{item}
+        Quadruple: $\quadruple{1}{2}{3}{4}$,
+      \end{item}
+      \begin{item}
+        Qintuple: $\quintuple{1}{2}{3}{4}{5}$,
+      \end{item}
+      \begin{item}
+        Sextuple: $\sextuple{1}{2}{3}{4}{5}{6}$,
+      \end{item}
+      \begin{item}
+        Septuple: $\septuple{1}{2}{3}{4}{5}{6}{7}$.
+      \end{item}
+    \end{itemize}
+    %
+    The factorial of the number $10$ is $\factorial{10}$.
 
     The Cartesian product of two sets $A$ and $B$ is
     $\cartprod{A}{B}$; if we take the product with $C$ as well, then
@@ -83,6 +112,7 @@
     \begin{equation*}
       \directsummany{k=1}{\infty}{V_{k}} \ne \cartprodmany{k=1}{\infty}{V_{k}}.
     \end{equation*}
+    %
     Here are a few common tuple spaces that should not have a
     superscript when that superscript would be one: $\Nn[1]$,
     $\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
@@ -101,7 +131,7 @@
     \begin{equation*}
       \unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}}
     \end{equation*}
-
+    %
     Finally, we have the four standard types of intervals in $\Rn[1]$,
     %
     \begin{align*}
@@ -142,7 +172,14 @@
   \end{section}
 
   \begin{section}{Font}
-    We can write things like Carathéodory and Güler and $\mathbb{R}$.
+    We can write things like Carathéodory and Güler and
+    $\mathbb{R}$. The PostScript Zapf Chancery font is also available
+    in both upper- and lower-case:
+    %
+    \begin{itemize}
+      \begin{item}$\mathpzc{abcdefghijklmnopqrstuvwxyz}$\end{item}
+      \begin{item}$\mathpzc{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$\end{item}
+    \end{itemize}
   \end{section}
 
   \begin{section}{Linear algebra}
@@ -153,7 +190,12 @@
     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
     $\transpose{L}$. Its trace is $\trace{L}$. Another matrix-specific
     concept is the Moore-Penrose pseudoinverse of $L$, denoted by
-    $\pseudoinverse{L}$.
+    $\pseudoinverse{L}$. Finally, the rank of a matrix $L$ is
+    $\rank{L}$. As far as matrix spaces go, we have the $n$-by-$n$
+    real-symmetric and complex-Hermitian matrices $\Sn$ and $\Hn$
+    respectively; however $\Sn[1]$ and $\Hn[1]$ do not automatically
+    simplify because the ``$n$'' does not indicate the arity of a
+    Cartesian product in this case.
 
     The span of a set $X$ is $\spanof{X}$, and its codimension is
     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
@@ -219,11 +261,6 @@
     system to test them.
   \end{section}
 
-  \begin{section}{Miscellaneous}
-    The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
-    = 3$.
-  \end{section}
-
   \begin{section}{Proof by cases}
 
     \begin{proposition}
@@ -270,6 +307,11 @@
     \renewcommand{\baselinestretch}{1}
   \end{section}
 
+  \begin{section}{Set theory}
+    The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
+    = 3$, and its powerset is $\powerset{X}$.
+  \end{section}
+
   \begin{section}{Theorems}
     \begin{corollary}
       The