% % Only the most commonly-used macros. Needed by everything else. % % Place the argument in matching left/right parntheses. \providecommand*{\of}[1]{ \left({#1}\right) } % Group terms using parentheses. \providecommand*{\qty}[1]{ \left({#1}\right) } % Group terms using square brackets. \providecommand*{\sqty}[1]{ \left[{#1}\right] } % Create a set from the given elements \providecommand*{\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. \providecommand*{\setc}[2]{\left\lbrace{#1}\ \middle|\ {#2} \right\rbrace} % A pair of things. \providecommand*{\pair}[2]{ \left({#1},{#2}\right) } % A triple of things. \providecommand*{\triple}[3]{ \left({#1},{#2},{#3}\right) } % The Cartesian product of two things. \providecommand*{\cartprod}[2]{ {#1}\times{#2} } % The Cartesian product of three things. \providecommand*{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} } % The direct sum of two things. \providecommand*{\directsum}[2]{ {#1}\oplus{#2} } % The factorial operator. \providecommand*{\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. \providecommand*{\Nn}[1][n]{ \mathbb{N}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi } % The integral n-space, Z x Z x Z x ... x Z. \providecommand*{\Zn}[1][n]{ \mathbb{Z}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi } % The rational n-space, Q x Q x Q x ... x Q. \providecommand*{\Qn}[1][n]{ \mathbb{Q}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi } % The real n-space, R x R x R x ... x R. \providecommand*{\Rn}[1][n]{ \mathbb{R}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi } % The complex n-space, C x C x C x ... x C. \providecommand*{\Cn}[1][n]{ \mathbb{C}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi } % % Basic set operations % % The union of its two arguments. \providecommand*{\union}[2]{ {#1} \cup {#2} } % A three-argument union. \providecommand*{\unionthree}[3]{ \union{\union{#1}{#2}}{#3} } % The intersection of its two arguments. \providecommand*{\intersect}[2]{ {#1} \cap {#2} } % A three-argument intersection. \providecommand*{\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. \providecommand*{\binopmany}[4]{ \mathchoice { \underset{#2}{\overset{#3}{#1}}{#4} } { {#1}_{#2}^{#3}{#4} } { {#1}_{#2}^{#3}{#4} } { {#1}_{#2}^{#3}{#4} } } \providecommand*{\unionmany}[3]{ \binopmany{\cup}{#1}{#2}{#3} } \providecommand*{\intersectmany}[3]{ \binopmany{\cap}{#1}{#2}{#3} }