mjo-set.tex

   1 %
   2 % Things that fit absolutely nowhere else.
   3 %
   4 \ifx\havemjoset\undefined
   5 \def\havemjoset{1}
   6
   7 \input{mjo-common} % binopmany
   8 \input{mjo-font} % amsfonts and \mathpzc
   9
  10 \ifx\operatorname\undefined
  11   \usepackage{amsopn}
  12 \fi
  13
  14 \ifx\bigtimes\undefined
  15   \usepackage{mathtools}
  16 \fi
  17
  18 % Create a set from the given elements
  19 \newcommand*{\set}[1]{\left\lbrace{#1}\right\rbrace}
  20
  21 % A set comprehension, where the ``such that...'' bar is added
  22 % automatically. The bar was chosen over a colon to avoid ambiguity
  23 % with the L : V -> V notation. We can't leverage \set here because \middle
  24 % needs \left and \right present.
  25 \newcommand*{\setc}[2]{\left\lbrace{#1}\ \middle|\ {#2} \right\rbrace}
  26
  27
  28 % The cardinality of a set. The |X| notation conflicts with the
  29 % absolute value, and the meaning of card(X) is clear at once, so we
  30 % prefer the latter.
  31 \newcommand*{\card}[1]{ \operatorname{card}\of{{#1}} }
  32
  33
  34 % The powerset of (that is, the set of all subsets of) its argument.
  35 \newcommand*{\powerset}[1]{\mathpzc{P}\of{{#1}}}
  36 \ifdefined\newglossaryentry
  37   \newglossaryentry{powerset}{
  38     name={\ensuremath{\powerset{X}}},
  39     description={the ``powerset,'' or set of all subsets of $X$},
  40     sort=p
  41   }
  42 \fi
  43
  44
  45 %
  46 % Basic set operations
  47 %
  48
  49 % The union of its two arguments.
  50 \newcommand*{\union}[2]{ {#1}\cup{#2} }
  51
  52 % A three-argument union.
  53 \newcommand*{\unionthree}[3]{ \union{\union{#1}{#2}}{#3} }
  54
  55 % The indexed union of many things.
  56 \newcommand*{\unionmany}[3]{ \binopmany{\bigcup}{#1}{#2}{#3} }
  57
  58 % The intersection of its two arguments.
  59 \newcommand*{\intersect}[2]{ {#1}\cap{#2} }
  60
  61 % A three-argument intersection.
  62 \newcommand*{\intersectthree}[3]{ \intersect{\intersect{#1}{#2}}{#3} }
  63
  64 % The indexed intersection of many things.
  65 \newcommand*{\intersectmany}[3]{ \binopmany{\bigcap}{#1}{#2}{#3} }
  66
  67 % The Cartesian product of two things.
  68 \newcommand*{\cartprod}[2]{ {#1}\times{#2} }
  69
  70 % The Cartesian product of three things.
  71 \newcommand*{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} }
  72
  73 % The indexed Cartesian product of many things.
  74 \newcommand*{\cartprodmany}[3]{ \binopmany{\bigtimes}{#1}{#2}{#3} }
  75
  76
  77 \fi