Steven has already said enough, but just for completeness and for proving that LaTeX offers always new roads I'll post another solution by means of .., though I think cases environment is the most appropriate for this job.
\documentclass{article}
\usepackage{amsmath}
\begin{document}
[
t(v) = \left{\begin{array}{lr}
\text{John} & \text{hvis $v$ er variabelen $x$}\
\text{Paul} & \text{hvis $v$ er variabelen $y$}\
\text{George} & \text{hvis $v$ er variabelen $z$}\
\text{Yoko} & \text{ellers}
\end{array}\right.
]
\end{document}
This produces

Notice that amsmath package is only needed for text command. A benefit from using array would be that you can control alignment by means of r and l (respectively right aligned, and left aligned) arguments in \begin{array}{lr}, and you can add lines as well as control manually spacing, by writing
\[
t(v) = \left\{\begin{array}{l@{\hspace{3em}}|r}
\text{John} & \text{hvis $v$ er variabelen $x$}\\
\hline
\text{Paul} & \text{hvis $v$ er variabelen $y$}\\
\text{George} & \text{hvis $v$ er variabelen $z$}\\
\text{Yoko} & \text{ellers}
\end{array}\right.
\]
@{\hspace{3em}} can be set to whichever value in em, cm, pt and other measure you like for making custom space between columns, while | can be used for adding vertical lines bewreen the arry columns. Finally adding \hline you can achieve horizontal rules.
Little addendum: \left\{ and \right. fake the case behaviour of cases environment, so that if you prefer you can switch it to othe parenthesis, as [, (, \langle and so on.