This paper explores association between the notions of similarity and preference by using the framework of the theory of binary relations considered as subsets of the cartesian product of the set of objects by itself. Unordered partitions correspond to the so-called equivalence relations, and ordered partitions, to the so-called weak order relations. We derive a number of properties of the metric space of equivalence and weak order relations. One of them is establishing of the fact that the so-called Kemeny distance between tied rankings is identical to the mismatch distance between corresponding binary relations of weak order.