Add the \preimage macro.

[mjotex.git] / examples.tex

diff --git a/examples.tex b/examples.tex
index c87f59bdc792d78d7d5e88e0158510616bf63b56..e0ffcb97a11ce5a97a2e837e7c994fd19714e1ae 100644 (file)
--- a/examples.tex
+++ b/examples.tex
@@ -28,7 +28,8 @@
   \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
-    $\inverse{f}$.
+    $\inverse{f}$. 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}
 
   \begin{section}{Common}
@@ -51,7 +52,16 @@
     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: $\intersect{A}{B}$,
-    $\intersectthree{A}{B}{C}$.
+    $\intersectthree{A}{B}{C}$. 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*}
+    %
   \end{section}
 
   \begin{section}{Cone}
@@ -94,6 +104,30 @@
     The set of all bounded linear operators from $V$ to $W$ is
     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
     instead.
+
+    The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
+    but what if $W = V^{\perp}$? Then we wish to indicate that fact by
+    writing $\directsumperp{V}{W}$. That operator should survive a
+    display equation, too, and the weight of the circle should match
+    that of the usual direct sum operator.
+    %
+    \begin{align*}
+      Z = \directsumperp{V}{W}\\
+      \oplus \oplusperp \oplus \oplusperp
+    \end{align*}
+    %
+    Its form should also survive in different font sizes...
+    \Large
+    \begin{align*}
+      Z = \directsumperp{V}{W}\\
+      \oplus \oplusperp \oplus \oplusperp
+    \end{align*}
+    \Huge
+    \begin{align*}
+      Z = \directsumperp{V}{W}\\
+      \oplus \oplusperp \oplus \oplusperp
+    \end{align*}
+    \normalsize
   \end{section}
 
   \begin{section}{Listing}