  An equivalence relation can be constructed from a given (homogeneous, binary)
relation in two steps: first, construct the smallest reflexive and transitive
relation containing the given relation (the "star" of the relation) and,
second, construct the largest symmetric relation that is included in the result
of the first step. The fact that the final result is also reflexive and
transitive (as well as symmetric), and thus an equivalence relation, is not
immediately obvious, although straightforward to prove. Rather than prove that
the defining properties of reflexivity and transitivity are satisfied, we
establish reflexivity and transitivity \emph{constructively} by exhibiting a
particular starth root -- in a way that emphasises the creative process in its
construction. The constructed starth root is fundamental to algorithms that
determinethe strongly connected components of a graph as well as the
decomposition of a graph into its strongly connected components together with
an acyclic graph connecting such components.
