\usepackage{amsmath}
%\usepackage{mathtools}
\usepackage{bbm}
\newcommand{\set}[1]{\left\lbrace #1 \right\rbrace }
\newcommand{\setZ}{\mathbbm{Z}}
\newcommand{\setN}{\mathbbm{N}}
\newcommand{\setQ}{\mathbbm{Q}}
\newcommand{\setR}{\mathbbm{R}}
\newcommand{\setC}{\mathbbm{C}}
\newcommand{\inlinematrix}[1]{\begin{pmatrix} #1 \end{pmatrix}}
\newcommand{\inlinematrixu}[1]{\begin{matrix} #1 \end{matrix}}
\newcommand{\inlinematrixd}[2]{\left( \begin{matrix} #1\vphantom{#2} \end{matrix} ~\right| \left. \begin{matrix} \vphantom{#1}#2 \end{matrix} \right)}
\newcommand{\inlinematrixdet}[1]{\begin{vmatrix} #1 \end{vmatrix}}

\newcommand{\norm}[1]{\left\Vert #1 \right\Vert}
\newcommand{\scalar}[2]{\langle #1 , #2 \rangle}
\newcommand{\abs}[1]{{\left| #1 \right|}}
\newcommand{\gdw}{\Leftrightarrow}
%\newcommand{\iff}{\Leftrightarrow}
\newcommand{\re}[1]{{\text{Re}\left( #1 \right)}}
\newcommand{\im}[1]{{\text{Im}\left( #1 \right)}}
\newcommand{\tr}{{\text{tr}}}

\newcommand{\QED}{\begin{large}\textbf{\checkmark}\end{large}}
\newcommand{\cequiv}{\stackrel{\scriptscriptstyle\wedge}{=}}

\newcommand{\diffP}[1]{\frac{\partial}{\partial #1}}
\newcommand{\diffT}[1]{\frac{\text{d}}{\text{d} #1}}
\newcommand{\diffPfrac}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\intgr}[4]{\int_{#1}^{#2} #3 ~\text{d}#4}
\newcommand{\intgru}[2]{\int #1 ~\text{d}#2}