Bigcup, bigwedge etc. inside brackets gets space above

Question

I want to have the following formula:

\[\Phi=\{\phi\}\cup\left\{\exists \mathsf{x_0}\ldots\exists \mathsf{x_{n-1}}\left[\bigwedge_{0\leq i<j\leq n-1} \mathsf{x_i}\neq\mathsf{x_j}\right]\right\}\]

which outputs to this:

As you can see, I get a huge space above my \bidwedge. The usage of \limits doesn't change anything. How can I get a more decent looking set?

Answer 1

\left and \right always produce fences which are symmetric with respect to the formula axis, that is, the imaginary line where fraction lines would lie on.

With big operators having limits there is no need the fences cover them; in this case \bigg is sufficient and avoids too large delimiters. There's only one small manual fix: adding \, between the bracket and the operator, because the limit is very wide.

\documentclass{article}
\usepackage{amsmath}
\begin{document}

\[
\Phi=\{\phi\}\cup
  \biggl\{\exists \mathsf{x_0}\dots
         \exists \mathsf{x_{n-1}}
  \biggl[\,\bigwedge_{0\leq i<j\leq n-1} \mathsf{x_i}\neq\mathsf{x_j}
  \biggr]
  \biggr\}
\]

\end{document}

Actually, I would give a try to \Big:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\[
\Phi=\{\phi\}\cup
  \Bigl\{\exists \mathsf{x_0}\dots
         \exists \mathsf{x_{n-1}}
  \Bigl[\,\bigwedge_{0\leq i<j\leq n-1} \mathsf{x_i}\neq\mathsf{x_j}
  \Bigr]
  \Bigr\}
\]
\end{document}

Answer 2

You can use one of the \big,\Big,\bigg, \Bigg commands to adjust the size of the delimiters to your needs. This can be automatised with the DeclarePairedDelimiter from mathtools. An example is the \set command which be used in the \set* version, which corresponds to a pair of \left … \right; alternatively, it accepts an optional argument \big, \Big, &c.

I suggest one of these codes, without square brackets, that don't look too good in this context, in my opinion:

\documentclass[12pt]{article}

\usepackage{mathtools}
\DeclarePairedDelimiter\set{\{}{\}}

\begin{document}

\[\Phi=\{\phi\}\cup\set[\bigg]{\exists \mathsf{x_0}\ldots\exists \mathsf{x_{n-1}}\left(\smash[b]{\bigwedge_{0\leq i<j\leq n-1}} \mathsf{x_i}\neq\mathsf{x_j}\right)}\]
\[\Phi=\{\phi\}\cup\biggl\{\exists \mathsf{x_0}\ldots\exists \mathsf{x_{n-1}}\textsf{, } \smash[b]{\bigwedge_{0\leq i<j\leq n-1}} \mathsf{x_i}\neq\mathsf{x_j}\biggr\}\]

\end{document} 

Answer 3

You may want to reserve the use of \mathsf for the names of variables; I wouldn't extend its use to the indexing variables 0, i, j, n-1, etc. To write the expression more compactly, you may also want to load the mathtools package (a superset of the amsmath package) and that package's \smashoperator directive. Finally, as the other answers have suggested as well, I wouldn't use the \left and \right directives to size the parentheses since the size that's produced by this method is excessive, typographically speaking; use \biggl and \biggr instead.

\documentclass{article}
\usepackage{mathtools} % for '\smashoperator' macro
\begin{document}
\[
\Phi=\{\phi\} \cup
     \biggl\{\exists \mathsf{x}_0 \dots \exists \mathsf{x}_{n-1}
     \biggl[ \, \smashoperator[r]{\bigwedge_{0\leq i<j\leq n-1}} 
     \mathsf{x}_i \neq \mathsf{x}_j
  \biggr]
  \biggr\}
\]
\end{document}