mjo-common.tex

   1 %
   2 % Only the most commonly-used macros. Needed by everything else.
   3 %
   4
   5 % Needed for \mathbb.
   6 \usepackage{amsfonts}
   7
   8 % Place the argument in matching left/right parntheses.
   9 \providecommand*{\of}[1]{ \left({#1}\right) }
  10
  11 % Group terms using parentheses.
  12 \providecommand*{\qty}[1]{ \left({#1}\right) }
  13
  14 % Group terms using square brackets.
  15 \providecommand*{\sqty}[1]{ \left[{#1}\right] }
  16
  17 % Create a set from the given elements
  18 \providecommand*{\set}[1]{\left\lbrace{#1}\right\rbrace}
  19
  20 % A set comprehension, where the ``such that...'' bar is added
  21 % automatically. The bar was chosen over a colon to avoid ambiguity
  22 % with the L : V -> V notation. We can't leverage \set here because \middle
  23 % needs \left and \right present.
  24 \providecommand*{\setc}[2]{\left\lbrace{#1}\ \middle|\ {#2} \right\rbrace}
  25
  26 % A pair of things.
  27 \providecommand*{\pair}[2]{ \left({#1},{#2}\right) }
  28
  29 % A triple of things.
  30 \providecommand*{\triple}[3]{ \left({#1},{#2},{#3}\right) }
  31
  32 % The Cartesian product of two things.
  33 \providecommand*{\cartprod}[2]{ {#1}\times{#2} }
  34
  35 % The Cartesian product of three things.
  36 \providecommand*{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} }
  37
  38 % The direct sum of two things.
  39 \providecommand*{\directsum}[2]{ {#1}\oplus{#2} }
  40
  41 % The direct sum of three things.
  42 \providecommand*{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
  43
  44 % The factorial operator.
  45 \providecommand*{\factorial}[1]{ {#1}! }
  46
  47 %
  48 % Product spaces
  49 %
  50 % All of the product spaces (for example, R^n) that follow default to
  51 % an exponent of ``n'', but that exponent can be changed by providing
  52 % it as an optional argument. If the exponent given is ``1'', then it
  53 % will be omitted entirely.
  54 %
  55
  56 % The natural n-space, N x N x N x ... x N.
  57 \providecommand*{\Nn}[1][n]{
  58   \mathbb{N}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  59 }
  60
  61 % The integral n-space, Z x Z x Z x ... x Z.
  62 \providecommand*{\Zn}[1][n]{
  63   \mathbb{Z}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  64 }
  65
  66 % The rational n-space, Q x Q x Q x ... x Q.
  67 \providecommand*{\Qn}[1][n]{
  68   \mathbb{Q}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  69 }
  70
  71 % The real n-space, R x R x R x ... x R.
  72 \providecommand*{\Rn}[1][n]{
  73   \mathbb{R}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  74 }
  75
  76 % The complex n-space, C x C x C x ... x C.
  77 \providecommand*{\Cn}[1][n]{
  78   \mathbb{C}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
  79 }
  80
  81
  82 %
  83 % Basic set operations
  84 %
  85
  86 % The union of its two arguments.
  87 \providecommand*{\union}[2]{ {#1} \cup {#2} }
  88
  89 % A three-argument union.
  90 \providecommand*{\unionthree}[3]{ \union{\union{#1}{#2}}{#3} }
  91
  92 % The intersection of its two arguments.
  93 \providecommand*{\intersect}[2]{ {#1} \cap {#2} }
  94
  95 % A three-argument intersection.
  96 \providecommand*{\intersectthree}[3]{ \intersect{\intersect{#1}{#2}}{#3} }
  97
  98 % An indexed arbitrary binary operation such as the union or
  99 % intersection of an infinite number of sets. The first argument is
 100 % the operator symbol to use, such as \cup for a union. The second
 101 % argument is the lower index, for example k=1. The third argument is
 102 % the upper index, such as \infty. Finally the fourth argument should
 103 % contain the things (e.g. indexed sets) to be operated on.
 104 \providecommand*{\binopmany}[4]{
 105   \mathchoice
 106   { \underset{#2}{\overset{#3}{#1}}{#4} }
 107   { {#1}_{#2}^{#3}{#4} }
 108   { {#1}_{#2}^{#3}{#4} }
 109   { {#1}_{#2}^{#3}{#4} }
 110 }
 111
 112 \providecommand*{\unionmany}[3]{ \binopmany{\cup}{#1}{#2}{#3} }
 113 \providecommand*{\intersectmany}[3]{ \binopmany{\cap}{#1}{#2}{#3} }