mjo-set: adopt \powerset{} from mjo-common.

[mjotex.git] / examples.tex

diff --git a/examples.tex b/examples.tex
index c00279b16beb44514ae8e80e371d8dc3f8392a35..e11815ef7856b47f0dd5cb0e775c3272caeec427 100644 (file)
--- 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
@@ -130,7 +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*}
@@ -171,7 +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}
@@ -182,7 +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
@@ -248,11 +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}
@@ -299,6 +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