% % Only the most commonly-used macros. Needed by everything else. % \ifx\havemjocommon\undefined \def\havemjocommon{1} \ifx\mathbb\undefined \usepackage{amsfonts} \fi \ifx\bigtimes\undefined \usepackage{mathtools} \fi % Place the argument in matching left/right parentheses. \newcommand*{\of}[1]{ \left({#1}\right) } % Group terms using parentheses. \newcommand*{\qty}[1]{ \left({#1}\right) } % Group terms using square brackets. \newcommand*{\sqty}[1]{ \left[{#1}\right] } % Create a set from the given elements \newcommand*{\set}[1]{\left\lbrace{#1}\right\rbrace} % A set comprehension, where the ``such that...'' bar is added % automatically. The bar was chosen over a colon to avoid ambiguity % with the L : V -> V notation. We can't leverage \set here because \middle % needs \left and \right present. \newcommand*{\setc}[2]{\left\lbrace{#1}\ \middle|\ {#2} \right\rbrace} % A pair of things. \newcommand*{\pair}[2]{ \left({#1},{#2}\right) } % A triple of things. \newcommand*{\triple}[3]{ \left({#1},{#2},{#3}\right) } % A four-tuple of things. \newcommand*{\quadruple}[4]{ \left({#1},{#2},{#3},{#4}\right) } % A five-tuple of things. \newcommand*{\quintuple}[5]{ \left({#1},{#2},{#3},{#4},{#5}\right) } % A six-tuple of things. \newcommand*{\sextuple}[6]{ \left({#1},{#2},{#3},{#4},{#5},{#6}\right) } % A seven-tuple of things. \newcommand*{\septuple}[7]{ \left({#1},{#2},{#3},{#4},{#5},{#6},{#7}\right) } % The Cartesian product of two things. \newcommand*{\cartprod}[2]{ {#1}\times{#2} } % The Cartesian product of three things. \newcommand*{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} } % The direct sum of two things. \newcommand*{\directsum}[2]{ {#1}\oplus{#2} } % The direct sum of three things. \newcommand*{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} } % The factorial operator. \newcommand*{\factorial}[1]{ {#1}! } % % Product spaces % % All of the product spaces (for example, R^n) that follow default to % an exponent of ``n'', but that exponent can be changed by providing % it as an optional argument. If the exponent given is ``1'', then it % will be omitted entirely. % % The natural n-space, N x N x N x ... x N. \newcommand*{\Nn}[1][n]{ \mathbb{N}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi } \ifdefined\newglossaryentry \newglossaryentry{N}{ name={\ensuremath{\Nn[1]}}, description={the set of natural numbers}, sort=N } \fi % The integral n-space, Z x Z x Z x ... x Z. \newcommand*{\Zn}[1][n]{ \mathbb{Z}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi } \ifdefined\newglossaryentry \newglossaryentry{Z}{ name={\ensuremath{\Zn[1]}}, description={the ring of integers}, sort=Z } \fi % The rational n-space, Q x Q x Q x ... x Q. \newcommand*{\Qn}[1][n]{ \mathbb{Q}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi } \ifdefined\newglossaryentry \newglossaryentry{Q}{ name={\ensuremath{\Qn[1]}}, description={the field of rational numbers}, sort=Q } \fi % The real n-space, R x R x R x ... x R. \newcommand*{\Rn}[1][n]{ \mathbb{R}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi } \ifdefined\newglossaryentry \newglossaryentry{R}{ name={\ensuremath{\Rn[1]}}, description={the field of real numbers}, sort=R } \fi % The complex n-space, C x C x C x ... x C. \newcommand*{\Cn}[1][n]{ \mathbb{C}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi } \ifdefined\newglossaryentry \newglossaryentry{C}{ name={\ensuremath{\Cn[1]}}, description={the field of complex numbers}, sort=C } \fi % % Basic set operations % % The union of its two arguments. \newcommand*{\union}[2]{ {#1}\cup{#2} } % A three-argument union. \newcommand*{\unionthree}[3]{ \union{\union{#1}{#2}}{#3} } % The intersection of its two arguments. \newcommand*{\intersect}[2]{ {#1}\cap{#2} } % A three-argument intersection. \newcommand*{\intersectthree}[3]{ \intersect{\intersect{#1}{#2}}{#3} } % An indexed arbitrary binary operation such as the union or % intersection of an infinite number of sets. The first argument is % the operator symbol to use, such as \cup for a union. The second % argument is the lower index, for example k=1. The third argument is % the upper index, such as \infty. Finally the fourth argument should % contain the things (e.g. indexed sets) to be operated on. \newcommand*{\binopmany}[4]{ \mathchoice{ \underset{#2}{\overset{#3}{#1}}{#4} } { {#1}_{#2}^{#3}{#4} } { {#1}_{#2}^{#3}{#4} } { {#1}_{#2}^{#3}{#4} } } \newcommand*{\intersectmany}[3]{ \binopmany{\bigcap}{#1}{#2}{#3} } \newcommand*{\cartprodmany}[3]{ \binopmany{\bigtimes}{#1}{#2}{#3} } \newcommand*{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} } \newcommand*{\unionmany}[3]{ \binopmany{\bigcup}{#1}{#2}{#3} } % The four standard (UNLESS YOU'RE FRENCH) types of intervals along % the real line. \newcommand*{\intervaloo}[2]{ \left({#1},{#2}\right) } % open-open \newcommand*{\intervaloc}[2]{ \left({#1},{#2}\right] } % open-closed \newcommand*{\intervalco}[2]{ \left[{#1},{#2}\right) } % closed-open \newcommand*{\intervalcc}[2]{ \left[{#1},{#2}\right] } % closed-closed \fi