X-Git-Url: http://gitweb.michael.orlitzky.com/?p=mjotex.git;a=blobdiff_plain;f=examples.tex;h=3a1d6c4f77004e7e71182c4f8bdd1754659430e8;hp=cd615de16cbab3fc8381aac6aeb4039d945f5601;hb=f9205a3b883c08499edfdd3f9d1a6170a6fc6755;hpb=123eb0365b7a38288f0cd80a0c70b203b8ff33fd

diff --git a/examples.tex b/examples.tex
index cd615de..3a1d6c4 100644
--- a/examples.tex
+++ b/examples.tex
@@ -73,9 +73,8 @@
 
   \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 $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The tuples go
-    up to seven, for now:
+    like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. The tuples
+    go up to seven, for now:
     %
     \begin{itemize}
       \begin{item}
@@ -100,40 +99,21 @@
     %
     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
-    we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$ and $W$
-    is $\directsum{V}{W}$. Or three things,
-    $\directsumthree{U}{V}{W}$. How about more things? Like
-    $\directsummany{k=1}{\infty}{V_{k}} \ne
-    \cartprodmany{k=1}{\infty}{V_{k}}$. Those direct sums and
-    cartesian products adapt nicely to display equations:
+    The direct sum of $V$ and $W$ is $\directsum{V}{W}$. Or three
+    things, $\directsumthree{U}{V}{W}$. How about more things? Like
+    $\directsummany{k=1}{\infty}{V_{k}}$. Those direct sums
+    adapt nicely to display equations:
     %
     \begin{equation*}
-      \directsummany{k=1}{\infty}{V_{k}} \ne \cartprodmany{k=1}{\infty}{V_{k}}.
+      \directsummany{k=1}{\infty}{V_{k}} \ne \emptyset.
     \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
     superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$,
-    $\Qn[2]$, $\Rn[2]$, $\Cn[2]$.
-
-    We also have a few basic set operations, for example the union of
-    two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
-    course with union comes intersection: $\intersect{A}{B}$,
-    $\intersectthree{A}{B}{C}$. We can also take an arbitrary
-    (indexed) union and intersections of things, like
-    $\unionmany{k=1}{\infty}{A_{k}}$ or
-    $\intersectmany{k=1}{\infty}{B_{k}}$. The best part about those
-    is that they do the right thing in a display equation:
-    %
-    \begin{equation*}
-      \unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}}
-    \end{equation*}
-    %
-    The powerset of $X$ displays nicely, as $\powerset{X}$. Finally,
-    we have the four standard types of intervals in $\Rn[1]$,
+    $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. Finally, we have the four standard
+    types of intervals in $\Rn[1]$,
     %
     \begin{align*}
       \intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
@@ -192,7 +172,11 @@
     $\transpose{L}$. Its trace is $\trace{L}$. Another matrix-specific
     concept is the Moore-Penrose pseudoinverse of $L$, denoted by
     $\pseudoinverse{L}$. Finally, the rank of a matrix $L$ is
-    $\rank{L}$.
+    $\rank{L}$. 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
@@ -258,11 +242,6 @@
     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}
@@ -309,6 +288,35 @@
     \renewcommand{\baselinestretch}{1}
   \end{section}
 
+  \begin{section}{Set theory}
+    Here's a set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The
+    cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X} =
+    3$, and its powerset is $\powerset{X}$.
+
+    We also have a few basic set operations, for example the union of
+    two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
+    course with union comes intersection: $\intersect{A}{B}$,
+    $\intersectthree{A}{B}{C}$. The Cartesian product of two sets $A$
+    and $B$ is there too: $\cartprod{A}{B}$. If we take the product
+    with $C$ as well, then we obtain $\cartprodthree{A}{B}{C}$.
+
+    We can also take an arbitrary (indexed) union, intersection, or
+    Cartesian product of things, like
+    $\unionmany{k=1}{\infty}{A_{k}}$,
+    $\intersectmany{k=1}{\infty}{B_{k}}$, or
+    $\cartprodmany{k=1}{\infty}{C_{k}}$. The best part about those is
+    that they do the right thing in a display equation:
+    %
+    \begin{equation*}
+      \unionmany{k=1}{\infty}{A_{k}}
+      \ne
+      \intersectmany{k=1}{\infty}{B_{k}}
+      \ne
+      \cartprodmany{k=1}{\infty}{C_{k}}.
+    \end{equation*}
+    %
+  \end{section}
+
   \begin{section}{Theorems}
     \begin{corollary}
       The