You can use \smash to hide height and depth of contents. In the context of the \Set command from the braket package you need to apply this to each half of the contents separately. In your particular example you will see no difference (with standard fonts) as the braces are already at their smallest size.

\documentclass{article}
\usepackage{braket}
\begin{document}
$\Set{A\in R|A\in Q}$
$\Set{\smash{A\in R}|\smash{A\in Q}}$
$\Set{A\in R^{2}|A\in Q}$
$\Set{\smash{A\in R^{2}}|\smash{A\in Q}}$
\end{document}
In general it is best to avoid auto sizing, cf. Is it ever bad to use \left and \right? . The mathtools package documentation provides a \Set command that can be manually sized, e.g. \Set[\bigg], with an auto sizing variant \Set*. Here is the version I usually use:

\documentclass{article}
\usepackage{mathtools}
\newcommand{\with}{\SetSymbol[\delimsize]}
\newcommand{\SetSymbol}[1][]{\nonscript\:#1\vert
\allowbreak\nonscript\:\mathopen{}}
\DeclarePairedDelimiterX{\Set}[1]{\lbrace}{\rbrace}{#1}
\begin{document}
\( \Set{A\in R \with A\in Q} \)
\( \Set{A\in R^{2} \with A\in Q} \)
\( \Set[\big]{A\in R^{2} \with A\in Q} \)
\( \Set*{A\in R^{2} \with A\in Q} \)
\end{document}
You write \with instead of the | in the braket notation. It has the advantage over braket that if you change your mind later and wish to have the separator as a colon instead of a vertical line, then you just adjust the definition of \with (or of \SetSymbol).
\mathstrut? – Sep 13 '19 at 05:58\smash? – Peter Grill Sep 13 '19 at 06:03\Set{\smash[b]{A\in R| A\in Q}}? but this produce some error for me!! – Sep 13 '19 at 06:08\smash{...}is the opposite, making its contents zero height and depth. In the braket package's command\Setyou would have to\smasheach half by itself. But in your example the braces are already the smallest braket will use. For me themathtoolsapproach to a set command is preferable. – Andrew Swann Sep 13 '19 at 06:10\smashhere write\Set{\smash{A\in R}|\smash{A\in Q}}– Andrew Swann Sep 13 '19 at 06:15|? Thank you. – Sep 13 '19 at 06:17mathtoolsdocumentation defines\Setwith its\Set*variant which is auto scaling – Andrew Swann Sep 13 '19 at 06:22\Setfrombraket. There's no reason to. – egreg Sep 13 '19 at 08:31