2 % Only the most commonly-used macros. Needed by everything else.
4 \ifx\havemjocommon\undefined
8 \input{mjo-font
} % amsfonts and \mathpzc
10 \ifx\bigtimes\undefined
11 \usepackage{mathtools
}
14 % Place the argument in matching left/right parentheses.
15 \newcommand*
{\of}[1]{ \left(
{#1}\right)
}
17 % Group terms using parentheses.
18 \newcommand*
{\qty}[1]{ \left(
{#1}\right)
}
20 % Group terms using square brackets.
21 \newcommand*
{\sqty}[1]{ \left[{#1}\right] }
23 % Create a set from the given elements
24 \newcommand*
{\set}[1]{\left\lbrace{#1}\right\rbrace}
26 % A set comprehension, where the ``such that...'' bar is added
27 % automatically. The bar was chosen over a colon to avoid ambiguity
28 % with the L : V -> V notation. We can't leverage \set here because \middle
29 % needs \left and \right present.
30 \newcommand*
{\setc}[2]{\left\lbrace{#1}\
\middle|\
{#2} \right\rbrace}
33 \newcommand*
{\pair}[2]{ \left(
{#1},
{#2}\right)
}
36 \newcommand*
{\triple}[3]{ \left(
{#1},
{#2},
{#3}\right)
}
38 % A four-tuple of things.
39 \newcommand*
{\quadruple}[4]{ \left(
{#1},
{#2},
{#3},
{#4}\right)
}
41 % A five-tuple of things.
42 \newcommand*
{\quintuple}[5]{ \left(
{#1},
{#2},
{#3},
{#4},
{#5}\right)
}
44 % A six-tuple of things.
45 \newcommand*
{\sextuple}[6]{ \left(
{#1},
{#2},
{#3},
{#4},
{#5},
{#6}\right)
}
47 % A seven-tuple of things.
48 \newcommand*
{\septuple}[7]{ \left(
{#1},
{#2},
{#3},
{#4},
{#5},
{#6},
{#7}\right)
}
50 % The Cartesian product of two things.
51 \newcommand*
{\cartprod}[2]{ {#1}\times{#2} }
53 % The Cartesian product of three things.
54 \newcommand*
{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} }
56 % The direct sum of two things.
57 \newcommand*
{\directsum}[2]{ {#1}\oplus{#2} }
59 % The direct sum of three things.
60 \newcommand*
{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
62 % The factorial operator.
63 \newcommand*
{\factorial}[1]{ {#1}!
}
68 % All of the product spaces (for example, R^n) that follow default to
69 % an exponent of ``n'', but that exponent can be changed by providing
70 % it as an optional argument. If the exponent given is ``1'', then it
71 % will be omitted entirely.
74 % The natural n-space, N x N x N x ... x N.
75 \newcommand*
{\Nn}[1][n
]{
76 \mathbb{N
}\if\detokenize{#1}\detokenize{1}{}\else^
{#1}\fi
79 \ifdefined\newglossaryentry
81 name=
{\ensuremath{\Nn[1]}},
82 description=
{the set of natural numbers
},
87 % The integral n-space, Z x Z x Z x ... x Z.
88 \newcommand*
{\Zn}[1][n
]{
89 \mathbb{Z
}\if\detokenize{#1}\detokenize{1}{}\else^
{#1}\fi
92 \ifdefined\newglossaryentry
94 name=
{\ensuremath{\Zn[1]}},
95 description=
{the ring of integers
},
100 % The rational n-space, Q x Q x Q x ... x Q.
101 \newcommand*
{\Qn}[1][n
]{
102 \mathbb{Q
}\if\detokenize{#1}\detokenize{1}{}\else^
{#1}\fi
105 \ifdefined\newglossaryentry
106 \newglossaryentry{Q
}{
107 name=
{\ensuremath{\Qn[1]}},
108 description=
{the field of rational numbers
},
113 % The real n-space, R x R x R x ... x R.
114 \newcommand*
{\Rn}[1][n
]{
115 \mathbb{R
}\if\detokenize{#1}\detokenize{1}{}\else^
{#1}\fi
118 \ifdefined\newglossaryentry
119 \newglossaryentry{R
}{
120 name=
{\ensuremath{\Rn[1]}},
121 description=
{the field of real numbers
},
127 % The complex n-space, C x C x C x ... x C.
128 \newcommand*
{\Cn}[1][n
]{
129 \mathbb{C
}\if\detokenize{#1}\detokenize{1}{}\else^
{#1}\fi
132 \ifdefined\newglossaryentry
133 \newglossaryentry{C
}{
134 name=
{\ensuremath{\Cn[1]}},
135 description=
{the field of complex numbers
},
141 % The space of real symmetric n-by-n matrices. Does not reduce to
142 % merely "S" when n=1 since S^{n} does not mean an n-fold cartesian
144 \newcommand*
{\Sn}[1][n
]{ \mathcal{S
}^
{#1} }
145 \ifdefined\newglossaryentry
146 \newglossaryentry{Sn
}{
147 name=
{\ensuremath{\Sn}},
148 description=
{the set of $n$-by-$n$ real symmetric matrices
},
153 % The space of complex Hermitian n-by-n matrices. Does not reduce to
154 % merely "H" when n=1 since H^{n} does not mean an n-fold cartesian
156 \newcommand*
{\Hn}[1][n
]{ \mathcal{H
}^
{#1} }
157 \ifdefined\newglossaryentry
158 \newglossaryentry{Hn
}{
159 name=
{\ensuremath{\Hn}},
160 description=
{the set of $n$-by-$n$ complex Hermitian matrices
},
167 % Basic set operations
170 % The union of its two arguments.
171 \newcommand*
{\union}[2]{ {#1}\cup{#2} }
173 % A three-argument union.
174 \newcommand*
{\unionthree}[3]{ \union{\union{#1}{#2}}{#3} }
176 % The intersection of its two arguments.
177 \newcommand*
{\intersect}[2]{ {#1}\cap{#2} }
179 % A three-argument intersection.
180 \newcommand*
{\intersectthree}[3]{ \intersect{\intersect{#1}{#2}}{#3} }
182 % An indexed arbitrary binary operation such as the union or
183 % intersection of an infinite number of sets. The first argument is
184 % the operator symbol to use, such as \cup for a union. The second
185 % argument is the lower index, for example k=1. The third argument is
186 % the upper index, such as \infty. Finally the fourth argument should
187 % contain the things (e.g. indexed sets) to be operated on.
188 \newcommand*
{\binopmany}[4]{
189 \mathchoice{ \underset{#2}{\overset{#3}{#1}}{#4} }
190 { {#1}_
{#2}^
{#3}{#4} }
191 { {#1}_
{#2}^
{#3}{#4} }
192 { {#1}_
{#2}^
{#3}{#4} }
195 \newcommand*
{\intersectmany}[3]{ \binopmany{\bigcap}{#1}{#2}{#3} }
196 \newcommand*
{\cartprodmany}[3]{ \binopmany{\bigtimes}{#1}{#2}{#3} }
197 \newcommand*
{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
198 \newcommand*
{\unionmany}[3]{ \binopmany{\bigcup}{#1}{#2}{#3} }
200 \newcommand*
{\powerset}[1]{\mathpzc{P
}\of{{#1}}}
201 \ifdefined\newglossaryentry
202 \newglossaryentry{powerset
}{
203 name=
{\ensuremath{\powerset{X
}}},
204 description=
{the ``powerset,'' or set of all subsets of $X$
},
209 % The four standard (UNLESS YOU'RE FRENCH) types of intervals along
211 \newcommand*
{\intervaloo}[2]{ \left(
{#1},
{#2}\right)
} % open-open
212 \newcommand*
{\intervaloc}[2]{ \left(
{#1},
{#2}\right] } % open-closed
213 \newcommand*
{\intervalco}[2]{ \left[{#1},
{#2}\right)
} % closed-open
214 \newcommand*
{\intervalcc}[2]{ \left[{#1},
{#2}\right] } % closed-closed