Align/Array equation - explain the options

Question

I have searched and found a solution to a problem I had here: Horizontal alignment, but now I have a follow-up question (and can't comment there because I have little reputation):

The code in the answer is

\usepackage{amsmath}
BEFORE
\begin{equation*}
\renewcommand\arraystretch{1.5} % mimic wider spacing of align environment
\begin{array}{@{} r@{}c@{}l @{}}
   V & \overset{\textnormal{def}}{=}     & R(\delta) \\
     & \overset{ \{ 1.2.3 \} }{=} & \delta(a) R(a)
\end{array}
\end{equation*}

and this looks nice. However, when I remove all the @{} things, it looks different:

AFTER
\begin{equation*}
\renewcommand\arraystretch{1.5}
\begin{array}{rcl}
   V & \overset{\textnormal{def}}{=}     & R(\delta) \\
     & \overset{ \{ 1.2.3 \} }{=} & \delta(a) R(a)
\end{array}
\end{equation*}

What do the @{} mean? Can you explain to me where the difference lies? I thought, only \begin{array}{rcl} would suffice to get the exact same output.

Picture:

Answer 1

It would be better to use an environment designed for aligning displayed equations such as align from amsmath (which would set the display in display math mode and put the right space around the =.

array is designed for matrices and sets its content in inline math mode and is not optimised for aligning a single equation.

\begin{array}{rcl}

means a three column table but each column has \tabcolsep space (padding) either side so there is 2\tabcolsep between each column. @{hello} replaces the tabcolsep by the supplied text so this would write hello in each row. so @{} has the effect of removing the 2\tabcolsep space.

So r@{}c@{}l gives tighter spacing but neither gives the same space as an = normally gets in infix position compare with

\begin{align*}
   V & \overset{\mathrm{def}}{=}      R(\delta) \\
     & \overset{ \{ 1.2.3 \} }{=}  \delta(a) R(a)
\end{align*}