15

How can I combine the symbols \wedge and \vee into one symbol, like \wedge\vee and \vee\wedge?

I am thinking of something like \gtrless but rotated by 90 degrees.

The purpose is similar to the \pm symbol, to indicate an alternative.

13

This requires some trial and error:

\documentclass{article}

\usepackage{amssymb,graphicx}

\newcommand{\veewedge}{%
  \mathbin{{\vee}\mkern-5mu{\wedge}}%
}
\newcommand{\wedgevee}{%
  \mathbin{{\wedge}\mkern-5mu{\vee}}%
}

\begin{document}

$A\veewedge B\wedgevee C_{\veewedge+\wedgevee}$

\end{document}

enter image description here

This instead doesn't require trial and error but is, of course, less efficient.

\documentclass{article}

\usepackage{amsmath,amssymb,graphicx}

\newcommand{\veewedge}{\veeorwedge{\vee\wedge}}
\newcommand{\wedgevee}{\veeorwedge{\wedge\vee}}

\makeatletter
\newcommand{\veeorwedge}[1]{%
  \mathbin{%
    \mathpalette\vee@or@wedge{#1}\relax
  }%
}
\newcommand{\vee@or@wedge}[2]{\vee@@or@@wedge#1#2}
\newcommand{\vee@@or@@wedge}[3]{%
  \sbox\z@{$\m@th#1#3$}%
  {#2}\kern-.5\wd\z@\mkern2mu\box\z@
}
\makeatother

\begin{document}

$A\veewedge B\wedgevee C_{\veewedge+\wedgevee}$

\end{document}

The separation is decided upon by changing the \mkern value.

enter image description here

2

In the first line, I just construct several possibilities inline. I tried to interpret your intent in the form of the macro \gtlt, shown on the 2nd line. I note that the \gtlt symbol has the same vertical height as the \pm, which may be relevant to your intent.

On the 3rd line, I provide an alternative that you didn't request, but which you may find more satisfactory.

\documentclass{article}
\usepackage{stackengine}
\stackMath
\def\gtlt{\mathrel{%
  \stackon[1pt]{\scriptscriptstyle\wedge}{\scriptscriptstyle\vee}}}
\begin{document}
$x \vee \wedge 
\stackon[1pt]{\scriptstyle\vee}{\scriptstyle\wedge} 
\stackon[1pt]{\scriptscriptstyle\wedge}{\scriptscriptstyle\vee} y \pm$

$ x \pm y \gtlt z$

$ x \pm y \mathrel{\wedge\kern-3pt\vee} z$
\end{document}

enter image description here

0
\documentclass{article}
\usepackage{amsmath}

\newcommand{\wedgevee}{\mathrel{\substack{\wedge\\\vee}}}
\newcommand{\veewedge}{\mathrel{\substack{\vee\\\wedge}}}

\begin{document}

    \[ \vee \veewedge \wedgevee \wedge \]
    \[ (x_1 \wedge x_2) \wedgevee (x_3 \vee x_4) \veewedge (x_5 \wedge x_6) \]

\end{document}

enter image description here

As to your edit of a rotated \gtrless symbol, use:

\rotatebox{90}{$\gtrless$}
0

Why not try what were you thinking? Maybe be it's not such a bad idea.

A simple rotated box with \lessgtr and \gtrless is not spaced properly nor scaled inside ^{} or _{}, but with a little tuning seem to work:

MWE

\documentclass{article}
\usepackage{amssymb, graphicx,amstext}
\def\VA{\mathop{\text{\rotatebox[origin=c]{90}{\resizebox{.7em}{1em}{$\lessgtr$}}}}}
\def\AV{\mathop{\text{\rotatebox[origin=c]{90}{\resizebox{.8em}{1em}{$\gtrless$}}}}}
\begin{document}

\[ A \VA B \AV C_{ \AV + \VA} \]

\end{document}

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