mjo-common.tex

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