X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=examples.tex;h=af11de8284796dd1125b0ee8443cd9b1a9c617ca;hb=83458c3f65b80a7897ced0553804b38f4872fcb6;hp=e1f375fdc68d219d144010732030850d31cd5869;hpb=66f91bd13a303f46822639420a20b5bffbe7a963;p=mjotex.git

diff --git a/examples.tex b/examples.tex
index e1f375f..af11de8 100644
--- a/examples.tex
+++ b/examples.tex
@@ -112,6 +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
@@ -130,8 +131,9 @@
     \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 $\Rn[1]$,
+    %
+    The powerset of $X$ displays nicely, as $\powerset{X}$. 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 },\\
@@ -190,7 +192,11 @@
     $\transpose{L}$. 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
@@ -256,11 +262,6 @@
     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}
@@ -307,6 +308,11 @@
     \renewcommand{\baselinestretch}{1}
   \end{section}
 
+  \begin{section}{Set theory}
+    The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
+    = 3$.
+  \end{section}
+
   \begin{section}{Theorems}
     \begin{corollary}
       The