2 % Only the most commonly-used macros. Needed by everything else.
4 \ifx\havemjocommon\undefined
12 \ifx\bigtimes\undefined
13 \usepackage{mathtools
}
16 % Place the argument in matching left/right parentheses.
17 \newcommand*
{\of}[1]{ \left(
{#1}\right)
}
19 % Group terms using parentheses.
20 \newcommand*
{\qty}[1]{ \left(
{#1}\right)
}
22 % Group terms using square brackets.
23 \newcommand*
{\sqty}[1]{ \left[{#1}\right] }
25 % Create a set from the given elements
26 \newcommand*
{\set}[1]{\left\lbrace{#1}\right\rbrace}
28 % A set comprehension, where the ``such that...'' bar is added
29 % automatically. The bar was chosen over a colon to avoid ambiguity
30 % with the L : V -> V notation. We can't leverage \set here because \middle
31 % needs \left and \right present.
32 \newcommand*
{\setc}[2]{\left\lbrace{#1}\
\middle|\
{#2} \right\rbrace}
35 \newcommand*
{\pair}[2]{ \left(
{#1},
{#2}\right)
}
38 \newcommand*
{\triple}[3]{ \left(
{#1},
{#2},
{#3}\right)
}
40 % The Cartesian product of two things.
41 \newcommand*
{\cartprod}[2]{ {#1}\times{#2} }
43 % The Cartesian product of three things.
44 \newcommand*
{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} }
46 % The direct sum of two things.
47 \newcommand*
{\directsum}[2]{ {#1}\oplus{#2} }
49 % The direct sum of three things.
50 \newcommand*
{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
52 % The factorial operator.
53 \newcommand*
{\factorial}[1]{ {#1}!
}
58 % All of the product spaces (for example, R^n) that follow default to
59 % an exponent of ``n'', but that exponent can be changed by providing
60 % it as an optional argument. If the exponent given is ``1'', then it
61 % will be omitted entirely.
64 % The natural n-space, N x N x N x ... x N.
65 \newcommand*
{\Nn}[1][n
]{
66 \mathbb{N
}\if\detokenize{#1}\detokenize{1}{}\else^
{#1}\fi
69 \ifdefined\newglossaryentry
71 name=
{\ensuremath{\Nn[1]}},
72 description=
{the set of natural numbers
},
77 % The integral n-space, Z x Z x Z x ... x Z.
78 \newcommand*
{\Zn}[1][n
]{
79 \mathbb{Z
}\if\detokenize{#1}\detokenize{1}{}\else^
{#1}\fi
82 \ifdefined\newglossaryentry
84 name=
{\ensuremath{\Zn[1]}},
85 description=
{the ring of integers
},
90 % The rational n-space, Q x Q x Q x ... x Q.
91 \newcommand*
{\Qn}[1][n
]{
92 \mathbb{Q
}\if\detokenize{#1}\detokenize{1}{}\else^
{#1}\fi
95 \ifdefined\newglossaryentry
97 name=
{\ensuremath{\Qn[1]}},
98 description=
{the field of rational numbers
},
103 % The real n-space, R x R x R x ... x R.
104 \newcommand*
{\Rn}[1][n
]{
105 \mathbb{R
}\if\detokenize{#1}\detokenize{1}{}\else^
{#1}\fi
108 \ifdefined\newglossaryentry
109 \newglossaryentry{R
}{
110 name=
{\ensuremath{\Rn[1]}},
111 description=
{the field of real numbers
},
117 % The complex n-space, C x C x C x ... x C.
118 \newcommand*
{\Cn}[1][n
]{
119 \mathbb{C
}\if\detokenize{#1}\detokenize{1}{}\else^
{#1}\fi
122 \ifdefined\newglossaryentry
123 \newglossaryentry{C
}{
124 name=
{\ensuremath{\Cn[1]}},
125 description=
{the field of complex numbers
},
131 % The space of real symmetric n-by-n matrices.
132 \newcommand*
{\Sn}[1][n
]{
133 \mathcal{S
}\if\detokenize{#1}\detokenize{1}{}\else^
{#1}\fi
136 \ifdefined\newglossaryentry
137 \newglossaryentry{Sn
}{
138 name=
{\ensuremath{\Sn}},
139 description=
{the set of $n$-by-$n$ real symmetric matrices
},
144 % The space of complex Hermitian n-by-n matrices.
145 \newcommand*
{\Hn}[1][n
]{
146 \mathcal{H
}\if\detokenize{#1}\detokenize{1}{}\else^
{#1}\fi
149 \ifdefined\newglossaryentry
150 \newglossaryentry{Hn
}{
151 name=
{\ensuremath{\Hn}},
152 description=
{the set of $n$-by-$n$ complex Hermitian matrices
},
158 % Basic set operations
161 % The union of its two arguments.
162 \newcommand*
{\union}[2]{ {#1}\cup{#2} }
164 % A three-argument union.
165 \newcommand*
{\unionthree}[3]{ \union{\union{#1}{#2}}{#3} }
167 % The intersection of its two arguments.
168 \newcommand*
{\intersect}[2]{ {#1}\cap{#2} }
170 % A three-argument intersection.
171 \newcommand*
{\intersectthree}[3]{ \intersect{\intersect{#1}{#2}}{#3} }
173 % An indexed arbitrary binary operation such as the union or
174 % intersection of an infinite number of sets. The first argument is
175 % the operator symbol to use, such as \cup for a union. The second
176 % argument is the lower index, for example k=1. The third argument is
177 % the upper index, such as \infty. Finally the fourth argument should
178 % contain the things (e.g. indexed sets) to be operated on.
179 \newcommand*
{\binopmany}[4]{
180 \mathchoice{ \underset{#2}{\overset{#3}{#1}}{#4} }
181 { {#1}_
{#2}^
{#3}{#4} }
182 { {#1}_
{#2}^
{#3}{#4} }
183 { {#1}_
{#2}^
{#3}{#4} }
186 \newcommand*
{\intersectmany}[3]{ \binopmany{\bigcap}{#1}{#2}{#3} }
187 \newcommand*
{\cartprodmany}[3]{ \binopmany{\bigtimes}{#1}{#2}{#3} }
188 \newcommand*
{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
189 \newcommand*
{\unionmany}[3]{ \binopmany{\bigcup}{#1}{#2}{#3} }
192 % The four standard (UNLESS YOU'RE FRENCH) types of intervals along
194 \newcommand*
{\intervaloo}[2]{ \left(
{#1},
{#2}\right)
} % open-open
195 \newcommand*
{\intervaloc}[2]{ \left(
{#1},
{#2}\right] } % open-closed
196 \newcommand*
{\intervalco}[2]{ \left[{#1},
{#2}\right)
} % closed-open
197 \newcommand*
{\intervalcc}[2]{ \left[{#1},
{#2}\right] } % closed-closed