

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%% macro in math styles %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newcommand{\TT}{ \mathbb{T} }
\newcommand{\PP}{ \mathbb{P} }
%\newcommand{\AA}{ \mathbb{A} }

\newcommand{\NN}{ \mathbb{N} }
\newcommand{\ZZ}{ \mathbb{Z} }
\newcommand{\QQ}{ \mathbb{Q} }
\newcommand{\RR}{ \mathbb{R} }
\newcommand{\CC}{ \mathbb{C} }
\newcommand{\HH}{ \mathbb{H} }
\newcommand{\OO}{ \mathbb{O} }

\newcommand{\ep}{ \varepsilon }
\newcommand{\ph}{ \varphi }


\newcommand{\CASE}[1]{ {\left\{\begin{array}{ll} #1 \end{array}\right.} }


\newcommand{\numm}{ \,\,\, \mathrm{if \,\, and \,\, only \,\, if} \,\,\, }
\newcommand{\num}{ \,\,\, \mathrm{if} \,\,\, }

\newcommand{\unique}{ ! }

\newcommand{\et}{ \,\,\, \mathrm{and} \,\,\, }
\newcommand{\ou}{ \,\,\, \mathrm{or} \,\,\, }
\newcommand{\aut}{ \,\,\, \mathrm{or} \,\,\, }

\newcommand{\then}{ \mbox{ then } }
\newcommand{\wehave}{ \mbox{we have } }
%\newcommand{\implies}{ \mbox{ implies } }

\newcommand{\is}{ \mbox{ is } }
\newcommand{\isnot}{ \mbox{ is not } }


%\newcommand{\lq}{\mbox{`}}
%\newcommand{\rq}{\mbox{'}}
\newcommand{\lqq}{\mbox{``}}
\newcommand{\rqq}{\mbox{''}}

\newcommand{\set}[2]{
\left\{ \! \begin{array}{c|c} \! #1 \! & \! #2 \! \end{array} \! \right\}
}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\pl}{ \prec }
\newcommand{\pleq}{ \preceq }
\newcommand{\pg}{ \succ }
\newcommand{\pgeq}{ \succeq }


\newcommand{\tl}{ \triangleleft }
\newcommand{\tg}{ \triangleright }

\newcommand{\Tl}{ \lhd }
\newcommand{\Tleq}{ \unlhd }
\newcommand{\Tg}{ \rhd }
\newcommand{\Tgeq}{ \unrhd }

%arrows

\newcommand{\arrLR}{ \leftrightarrow }
\newcommand{\arrL}{ \leftarrow }
\newcommand{\arrR}{ \rightarrow }
\newcommand{\ArrLR}{ \Leftrightarrow }
\newcommand{\ArrL}{ \Leftarrow }
\newcommand{\ArrR}{ \Rightarrow }
\newcommand{\longarrLR}{ \longleftrightarrow }
\newcommand{\longarrL}{ \longleftarrow }
\newcommand{\longarrR}{ \longrightarrow }
\newcommand{\LongarrLR}{ \Longleftrightarrow }
\newcommand{\LongarrL}{ \Longleftarrow }
\newcommand{\LongarrR}{ \Longrightarrow }

\newcommand{\arrUD}{ \updownarrow }
\newcommand{\arrU}{ \uparrow }
\newcommand{\arrD}{ \downarrow }
\newcommand{\ArrUD}{ \Updownarrow }
\newcommand{\ArrU}{ \Uparrow }
\newcommand{\ArrD}{ \Downarrow }
\newcommand{\longarrUD}{ \updownarrow }
\newcommand{\longarrU}{ \uparrow }
\newcommand{\longarrD}{ \downarrow }
\newcommand{\LongarrUD}{ \Updownarrow }
\newcommand{\LongarrU}{ \Uparrow }
\newcommand{\LongarrD}{ \Downarrow }

\newcommand{\arrUR}{ \nearrow }
\newcommand{\arrDR}{ \searrow }
\newcommand{\arrDL}{ \swarrow }
\newcommand{\arrUL}{ \nwarrow }
\newcommand{\ArrUR}{ \Nearrow }
\newcommand{\ArrDR}{ \Searrow }
\newcommand{\ArrDL}{ \Swarrow }
\newcommand{\ArrUL}{ \Nwarrow }


% relation

\newcommand{\lsim}{ \lesssim }
\newcommand{\gsim}{ \gtrsim }
\newcommand{\simm}{ \approx }
\newcommand{\lsimm}{ \lessapprox }
\newcommand{\gsimm}{ \gtrapprox }

\newcommand{\simeqq}{ \cong }


% map

\newcommand{\id}[1]{ \mathrm{id}_{ #1 } }
\renewcommand{\Mapsto}{\rightarrow}            %%%% NOTE %%%%
\newcommand{\inj}{ \hookrightarrow }
\newcommand{\surj}{ \twoheadrightarrow }

\newcommand{\inU}{\setlength{\unitlength}{1.75mm}
  \begin{picture}(1,1)
  \put(0.5,0.875){\oval(1,1.5)[b]}
  \put(0.5,0.875){\line(0,-1){0.75}}
  \end{picture}
}
\newcommand{\maps}[4]{\!\!
  \mbox{
  \begin{tabular}[t]{ccc}
  \( #1 \) & \!\( \Mapsto \)\! & \( #4 \) \\
  \( \inU \) &               & \( \inU \) \\
  \( #2 \) & \!\( \mapsto \)\! & \( #3 \)
  \end{tabular}}
}







\newcommand{\plus}{\hspace{2.24mm}}%{+}
\newcommand{\nl}{\plus}




\newcommand{\st}{ \mbox{ s.t. } }


\newcommand{\resp}{ \mathrm{resp.}\, }
\newcommand{\ie}{ \,\,\, \mathrm{i.e.} \,\,\, }


\newcommand{\widecheck}[1]{ \widehat{ #1 } }
\newcommand{\widebar}[1]{ \overline{ #1 } }
\newcommand{\widevec}[1]{ \overline{ #1 } }

\newcommand{\overvec}[1]{ \stackrel{ \rightarrow }{ #1 } }
\newcommand{\overcev}[1]{ \stackrel{ \leftarrow }{ #1 } }
\newcommand{\overconj}[1]{ \bar{ #1 } }



