$$ %---- MACROS FOR SETS ----%

ewcommand{\znz}[1]{\mathbb{Z} / #1 \mathbb{Z}}

ewcommand{\twoheadrightarrowtail}{\mapsto\mathrel{\mspace{-15mu}}\rightarrow} % popular set names

ewcommand{\N}{\mathbb{N}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\I}{\mathbb{I}} % popular vector space notation

ewcommand{\V}{\mathbb{V}}

ewcommand{\W}{\mathbb{W}}

ewcommand{\B}{\mathbb{B}}

ewcommand{\D}{\mathbb{D}} %---- MACROS FOR FUNCTIONS ----% % linear algebra

ewcommand{\T}{\mathrm{T}} \renewcommand{\ker}{\mathrm{ker}}

ewcommand{\range}{\mathrm{range}} \renewcommand{\span}{\mathrm{span}}

ewcommand{\rref}{\mathrm{rref}} \renewcommand{\dim}{\mathrm{dim}}

ewcommand{\col}{\mathrm{col}}

ewcommand{

ullspace}{\mathrm{null}}

ewcommand{\row}{\mathrm{row}}

ewcommand{\rank}{\mathrm{rank}}

ewcommand{

ullity}{\mathrm{nullity}} \renewcommand{\det}{\mathrm{det}}

ewcommand{\proj}{\mathrm{proj}} \renewcommand{\H}{\mathrm{H}}

ewcommand{\trace}{\mathrm{trace}}

ewcommand{\diag}{\mathrm{diag}}

ewcommand{\card}{\mathrm{card}}

ewcommand

orm[1]{\left\lVert#1\right\rVert} % differential equations

ewcommand{\laplace}[1]{\mathcal{L}\{#1\}}

ewcommand{\F}{\mathrm{F}} % misc

ewcommand{\sign}{\mathrm{sign}}

ewcommand{\softmax}{\mathrm{softmax}} \renewcommand{\th}{\mathrm{th}}

ewcommand{\adj}{\mathrm{adj}}

ewcommand{\hyp}{\mathrm{hyp}} \renewcommand{\max}{\mathrm{max}} \renewcommand{\min}{\mathrm{min}}

ewcommand{\where}{\mathrm{\ where\ }}

ewcommand{\abs}[1]{\vert #1 \vert}

ewcommand{\bigabs}[1]{\big\vert #1 \big\vert}

ewcommand{\biggerabs}[1]{\Bigg\vert #1 \Bigg\vert}

ewcommand{\equivalent}{\equiv}

ewcommand{\cross}{\times} % statistics

ewcommand{\cov}{\mathrm{cov}}

ewcommand{\var}{\mathrm{var}}

ewcommand{\bias}{\mathrm{bias}}

ewcommand{\E}{\mathrm{E}}

ewcommand{\prob}{\mathrm{prob}}

ewcommand{\unif}{\mathrm{unif}}

ewcommand{\invNorm}{\mathrm{invNorm}}

ewcommand{\invT}{\mathrm{invT}}

ewcommand{\P}{\text{P}}

ewcommand{\pmf}{\text{pmf}}

ewcommand{\pdf}{\text{pdf}} % real analysis \renewcommand{\sup}{\mathrm{sup}} \renewcommand{\inf}{\mathrm{inf}} %---- MACROS FOR ALIASES AND REFORMATTING ----% % logic

ewcommand{\forevery}{\ \forall\ }

ewcommand{\OR}{\lor}

ewcommand{\AND}{\land}

ewcommand{\then}{\implies} % set theory

ewcommand{\impropersubset}{\subseteq}

ewcommand{

otimpropersubset}{

subseteq}

ewcommand{\propersubset}{\subset}

ewcommand{

otpropersubset}{

ot\subset}

ewcommand{\union}{\cup}

ewcommand{\Union}[2]{\bigcup\limits_{#1}^{#2}}

ewcommand{\intersect}{\cap}

ewcommand{\Intersect}[2]{\bigcap\limits_{#1}^{#2}}

ewcommand{\intersection}[2]{\bigcap\limits_{#1}^{#2}}

ewcommand{\Intersection}[2]{\bigcap\limits_{#1}^{#2}}

ewcommand{\closure}{\overline}

ewcommand{\compose}{\circ} % linear algebra

ewcommand{\subspace}{\le}

ewcommand{\angles}[1]{\langle #1 \rangle}

ewcommand{\identity}{\mathbb{1}}

ewcommand{\orthogonal}{\perp} \renewcommand{\parallel}[1]{#1^{||}} % calculus

ewcommand{\integral}[2]{\int\limits_{#1}^{#2}}

ewcommand{\limit}[1]{\lim\limits_{#1}}

ewcommand{\approaches}{\rightarrow} \renewcommand{\to}{\rightarrow}

ewcommand{\convergesto}{\rightarrow} % algebra

ewcommand{\summation}[2]{\sum

olimits_{#1}^{#2}}

ewcommand{\product}[2]{\prod\limits_{#1}^{#2}}

ewcommand{\by}{\times}

ewcommand{\integral}[2]{\int_{#1}^{#2}}

ewcommand{\ln}{\text{ln}} % exists commands

ewcommand{

otexist}{

exists\ }

ewcommand{\existsatleastone}{\exists\ }

ewcommand{\existsonlyone}{\exists!}

ewcommand{\existsunique}{\exists!} \let\oldexists\exists \renewcommand{\exists}{\ \oldexists\ } % statistics

ewcommand{\distributed}{\sim}

ewcommand{\onetoonecorresp}{\sim}

ewcommand{\independent}{\perp\!\!\!\perp}

ewcommand{\conditionedon}{\ |\ }

ewcommand{\given}{\ |\ }

ewcommand{

otg}{

gtr}

ewcommand{\yhat}{\hat{y}}

ewcommand{\betahat}{\hat{\beta}}

ewcommand{\sigmahat}{\hat{\sigma}}

ewcommand{\muhat}{\hat{\mu}}

ewcommand{\transmatrix}{\mathrm{P}} \renewcommand{\choose}{\binom} % misc

ewcommand{\infinity}{\infty} \renewcommand{\bold}{\textbf}

ewcommand{\italics}{\textit}

ewcommand{\step}{\text{step}} $$