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I would like to change the size $\bigvee$ so that the lowest point of the character is at the baseline and the character is no taller than an upper case character, say $T$.

\documentclass[paper=letter,fontsize=12pt]{scrreprt}
    \usepackage[utf8]{inputenc}
    \usepackage{amsmath}
    \usepackage{relsize}
    \begin{document}
    This is a textstyle big wedge $\textstyle{\bigwedge} T$.
    \end{document}

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The ordinary commands \wedge and \vee may suit your needs:

$T\wedge T\vee T$

enter image description here

But if you want to resize to match a T, you can use the scalerel package:

enter image description here

\documentclass[paper=letter,fontsize=12pt]{scrreprt}
\usepackage[utf8]{inputenc}   
\usepackage{amsmath,scalerel}

\newcommand{\newwedge}{\mathbin{\scalerel*{\bigwedge}{T}}}
\newcommand{\newvee}{\mathbin{\scalerel*{\bigvee}{T}}}

\begin{document}
$T\newwedge T$

$\newwedge T$

$\newwedge$

$A_{S\newwedge T_{X\newwedge Y}}$ 

\end{document}

As an alternative, you could scale the \vee command which gives a slightly different shape:

enter image description here

\newcommand{\newwedge}{\mathbin{\scalerel*{\wedge}{T}}}
\newcommand{\newvee}{\mathbin{\scalerel*{\vee}{T}}}
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    One reason to prefer ordinary \wedge is that the rescaled \bigwedge ends up (probably inevitably) with a lighter-than normal line weight. Apr 13, 2022 at 17:07
  • @Sandy G Using overleaf to compile $\wedge T\vee T$ the $\wedge$ and $\vee$ are the height of lower case characters. My question was poorly worded. I would like the join character to be the height of $T$ but stand-alone without the $T$. I tried converting your solution to a function with one variable but that didn't work.
    – Jay
    Apr 14, 2022 at 12:12
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    @Jay: Relative sizes of characters will depend on the font you're using. I made some edits. Does this do what you want? You can use \newwedge and \newvee anywhere in math mode, as stand-alone characters or as binary operations.
    – Sandy G
    Apr 14, 2022 at 12:51

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