%
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*}
- %
- 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 },\\
\begin{section}{Set theory}
The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
= 3$, and its powerset is $\powerset{X}$.
+
+ 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}
\usepackage{amsfonts}
\fi
-\ifx\bigtimes\undefined
- \usepackage{mathtools}
-\fi
-
% Place the argument in matching left/right parentheses.
\newcommand*{\of}[1]{ \left({#1}\right) }
% A seven-tuple of things.
\newcommand*{\septuple}[7]{ \left({#1},{#2},{#3},{#4},{#5},{#6},{#7}\right) }
-% The Cartesian product of two things.
-\newcommand*{\cartprod}[2]{ {#1}\times{#2} }
-
-% The Cartesian product of three things.
-\newcommand*{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} }
-
% The direct sum of two things.
\newcommand*{\directsum}[2]{ {#1}\oplus{#2} }
\fi
-%
-% Basic set operations
-%
-
-% The union of its two arguments.
-\newcommand*{\union}[2]{ {#1}\cup{#2} }
-
-% A three-argument union.
-\newcommand*{\unionthree}[3]{ \union{\union{#1}{#2}}{#3} }
-
-% The intersection of its two arguments.
-\newcommand*{\intersect}[2]{ {#1}\cap{#2} }
-
-% A three-argument intersection.
-\newcommand*{\intersectthree}[3]{ \intersect{\intersect{#1}{#2}}{#3} }
-
% An indexed arbitrary binary operation such as the union or
% intersection of an infinite number of sets. The first argument is
% the operator symbol to use, such as \cup for a union. The second
{ {#1}_{#2}^{#3}{#4} }
}
-\newcommand*{\intersectmany}[3]{ \binopmany{\bigcap}{#1}{#2}{#3} }
-\newcommand*{\cartprodmany}[3]{ \binopmany{\bigtimes}{#1}{#2}{#3} }
+
\newcommand*{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
-\newcommand*{\unionmany}[3]{ \binopmany{\bigcup}{#1}{#2}{#3} }
+
% The four standard (UNLESS YOU'RE FRENCH) types of intervals along
% the real line.
\ifx\havemjoset\undefined
\def\havemjoset{1}
+\input{mjo-common} % binopmany
\input{mjo-font} % amsfonts and \mathpzc
\ifx\operatorname\undefined
\usepackage{amsopn}
\fi
+\ifx\bigtimes\undefined
+ \usepackage{mathtools}
+\fi
+
+
% The cardinality of a set. The |X| notation conflicts with the
% absolute value, and the meaning of card(X) is clear at once, so we
% prefer the latter.
\newcommand*{\card}[1]{ \operatorname{card}\of{{#1}} }
+% The powerset of (that is, the set of all subsets of) its argument.
\newcommand*{\powerset}[1]{\mathpzc{P}\of{{#1}}}
\ifdefined\newglossaryentry
\newglossaryentry{powerset}{
\fi
+%
+% Basic set operations
+%
+
+% The union of its two arguments.
+\newcommand*{\union}[2]{ {#1}\cup{#2} }
+
+% A three-argument union.
+\newcommand*{\unionthree}[3]{ \union{\union{#1}{#2}}{#3} }
+
+% The indexed union of many things.
+\newcommand*{\unionmany}[3]{ \binopmany{\bigcup}{#1}{#2}{#3} }
+
+% The intersection of its two arguments.
+\newcommand*{\intersect}[2]{ {#1}\cap{#2} }
+
+% A three-argument intersection.
+\newcommand*{\intersectthree}[3]{ \intersect{\intersect{#1}{#2}}{#3} }
+
+% The indexed intersection of many things.
+\newcommand*{\intersectmany}[3]{ \binopmany{\bigcap}{#1}{#2}{#3} }
+
+% The Cartesian product of two things.
+\newcommand*{\cartprod}[2]{ {#1}\times{#2} }
+
+% The Cartesian product of three things.
+\newcommand*{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} }
+
+% The indexed Cartesian product of many things.
+\newcommand*{\cartprodmany}[3]{ \binopmany{\bigtimes}{#1}{#2}{#3} }
+
+
\fi
projects / mjotex.git / commitdiff