  In this paper, we offer a presentation for the Weyl group of an affine
reflection system $R$ of type $A_1$ as well as a presentation for the so called
hyperbolic Weyl group associated with an affine reflection system of type
$A_1$. Applying these presentations to extended affine Weyl groups, and using a
description of the center of the hyperbolic Weyl group, we also give a new
finite presentation for an extended affine Weyl group of type $A_1$. Our
presentation for the (hyperbolic) Weyl group of an affine reflection system of
type $A_1$ is the first non-trivial presentation given in such a generality,
and can be considered as a model for other types.
