math environment again

Question

This follows a previous post which can be found here.
I'm now trying to used that command but I get lots of compilation mistakes and I really don't understand where the problem comes from. This is how I'm using the command:

\documentclass{article}
\usepackage{mathtools}
\newcommand{\twopartdef}[3]{%
   \left\{
      \begin{array}{ll}
          #1 & \mbox{if } #2 \\
          #3 & \mbox{otherwise}
      \end{array}
   \right.
}
\begin{document}
 %some text ...
 \[
 join^\sharp_{KSet_n}(\{ \alpha_1, \hdots, \alpha_p \}, \{ \beta_1, \hdots, \beta_q \}) = 
\twopartdef{\{\alpha_1, \hdots, \alpha_p\} \cup \{\beta_1, \hdots, \beta_q\}}{p \leq n et q \leq n}{\top} 
\]
 %more text
\end{document}

Here are the main compilation errors:

Missing $ inserted
I've inserted a begin-math/end-math symbol since I think you left one out
....
I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'..
....
I can't figure out why you would want to use a tab mark here.
....
A left brace was mandatory here, so I've put one in.
....

I found that the problem is from this line (by commenting it)

\twopartdef{\{\alpha_1, \hdots, \alpha_p\} \cup \{\beta_1, \hdots, \beta_q\}}{p \leq n et q \leq n}{\top}

So I tried to put $ $, lbrace rbrace instead of \{ \} but it didn't change anything. I've used the same command for other definition and it work fine. What am I missing?

Edit: I tried compiling the code above alone and it worked. So why doesn't it work with the remaining document?

Answer 1

Your definition of \twopartdef makes for a very heavy syntax.

\documentclass{article}
\usepackage{mathtools}

\newcommand{\case}[1]{\text{if $#1$}}
\newcommand{\otherwise}{\text{otherwise}}
\newenvironment{multipartdef}
  {\begin{cases}}
  {\end{cases}}

\DeclareMathOperator{\join}{join}

\begin{document}
 %some text ...
 \[
\join^\sharp_{\mathit{KSet}_n}(\{ \alpha_1, \dots, \alpha_p \}, \{ \beta_1, \dots, \beta_q \}) = 
\begin{multipartdef}
\{\alpha_1, \hdots, \alpha_p\} \cup \{\beta_1, \hdots, \beta_q\} &
  \case{p \leq n \text{ and } q \leq n}
\\
\top & \otherwise
\end{multipartdef}
\]
 %more text
\end{document}

If you have more cases, you won't need different commands, but only to add lines to the environment. You have also less chances to get errors.

Answer 2

I assume you are using this definition for \twopartdef:

\usepackage{amsmath}
\newcommand{\twopartdef}[3]{%
  \begin{cases}
    #1 & \text{if #2} \\
    #3 & \text{otherwise}
  \end{cases}
}

this definition treats the second argument in text mode, so you need to write

$p \leq n$ et $q \leq n$

in the second argument of the command in your code (otherwise, the firs \leq will appear in text mode and this will trigger the error message).

Your corrected code:

\documentclass{article}
\usepackage{amsmath}
\newcommand{\twopartdef}[3]{%
  \begin{cases}
    #1 & \text{if #2} \\
    #3 & \text{otherwise}
  \end{cases}
}
\DeclareMathOperator{\join}{join}

\begin{document}

\[
 \join^\sharp_{KSet_n}(\{ \alpha_1, \hdots, \alpha_p \}, \{ \beta_1, \hdots, \beta_q \}) = 
\twopartdef{\{\alpha_1, \hdots, \alpha_p\} \cup \{\beta_1, \hdots, \beta_q\}}{$p \leq n$ et $q \leq n$}{\top} 
\]

\end{document}

And the result:

Notice that I used \DeclareMathOperator to define the operator \join that admits super/subscripts (in a similar fashion to \sum).