# Peano's dot notation

I am wondering what is the best way to typeset Peano's dot notation.

It is a subset symbol with dots on both sides. For example, for 4/3 I write: ::\supset :\!.. However the spacing and the size of the dots are not correct.

• \supset is by default defined as a relation, whereas in the scanned image it appears to be "ordinary". If you wrap {subset} in braces, it will treated as ordinary, without the spaces. – barbara beeton Oct 30 '19 at 15:25
• So maybe should I wrap subset, adding the points and make the all a new relation with mathtools or something. Thanks. – user152057 Oct 30 '19 at 15:26
• I don't know of anything in mathtools that directly addresses this, but, indeed, defining a new relation is a good idea. – barbara beeton Oct 30 '19 at 15:29

The period is slightly bigger than the dots in the colon (at least in the Computer Modern fonts, other fonts may differ).

You should also center the colon with respect to the math axis.

\documentclass{article}
\usepackage{amsmath}
\usepackage{bm}
\usepackage{graphicx}

% Peano colon
\newcommand{\pc}{{\mspace{-1mu}\mathop:\mspace{-1mu}}}
% Peano period
\newcommand{\pp}{{%
\mspace{-1mu}%
\sbox0{$:$}\sbox2{$\mathop:$}%
\raisebox{\dimexpr\ht2-\ht0}{\scalebox{0.99}{$.$}}%
\mspace{-1mu}%
}}
% Peano implies
\newcommand{\pim}{{\mspace{-1mu}\bm{\supset}\mspace{-1mu}}}

\begin{document}

$d(x,y) \mathrel{\pc\pc\pim\pc\pp} D$

\end{document}


• Thé .97 seems a bit arbitrary but it is a very neat solution. – user152057 Oct 30 '19 at 23:09
• I am sure of the your excellent answer. – Sebastiano Oct 30 '19 at 23:11
• @Blincer I changed it into 0.99 – egreg Oct 30 '19 at 23:13
• I found a cleaner solution by cutting the colon in half to get the point. You can check my answer. – user152057 Oct 30 '19 at 23:49

Improving egreg's answer in order to remove arbitrariness and to simplify the code, I finally get:

\documentclass[preview]{standalone}

% Defines specific colon, period and implies
\newcommand{\pc}{{:}}
\newcommand{\pp}{\adjustbox{clip,trim=0pt 0pt 0pt {.5\totalheight}}{$\pc$}}
\newcommand{\pim}{{\boldsymbol{\supset}}}

\begin{document}
$P \mathrel{\pp\pim\pc\pp} Q$
\end{document}


Here there is my humble solution. The output, for my opinion, seem very nice the distance of the dots (they are the same), as from this example, using smallmatrix with amsmath package. The dots are also adapted to the supset symbol.

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}

\begin{document}
$d(x,a)<r\begin{smallmatrix} .&.\\ .&. \end{smallmatrix}\mkern-7mu\supset \mkern-7mu\begin{smallmatrix} .&\\ .&. \end{smallmatrix}d(x,a)>r$
\end{document}

• I upvote without hesitation. I think I'll try to create a package to get a command \peano{left}{right} but I'm clearly inspired by this method, thank you. – user152057 Oct 30 '19 at 22:30
• @Blincer I am very happy when others are happy. I was very happy to help you too and I thank you. There are many great users who will surely give you a better answer than mine. – Sebastiano Oct 30 '19 at 22:32
• It's a smart trick! I just think that defining amathrel will be cleaner. – user152057 Oct 30 '19 at 22:34