Philosophically relations are interesting because of the historic prejudice, given its most forceful expression by Leibniz, that they are somehow ‘unreal’ compared to the intrinsic, monadic properties of things. A way of putting the idea is that if all the monadic properties of the objects of a domain are fixed, then the relational properties are fixed as well (relations supervene on monadic properties). But in modern logic and science there is no justification for this claim.
The central notions in the logical treatment of relations are as follows. The domain of a relation is the set of things that bear the relation to something. The range is the set of things that have something bear the relation to them. The field of a relation is the set of things that belong either to its domain, or to its range. A binary relation relates one element from its domain to one of its range: Rxy. Relations may be defined over greater numbers of things: for example, we can define Rxyz to be the relation holding between three numbers when x+y=z, and so up to relations defined over n-tuples of any size. For formal and mathematical purposes a relation may be identified with the class of ordered pairs (or in general ordered n-tuples) that satisfy it. So ‘father of’ becomes the set of all pairs x,y, such that x is the father of y, ‘is greater than’ becomes the set of all pairs x,y, such that x is greater than y, and so on. The main properties to be noticed in the theory of relations are indexed under their own headings: see antisymmetric, asymmetric, equivalence relation, irreflexive, ordering relation, reflexive, symmetric, transitive.