Relations

Abstract

We often have to say, of two objects x, y, that they are ‘related’, in some way. For example, if x, y∈ ℝ we might have ‘x < y’; if x, y are integers we might have ‘x - y is exactly divisible by 3’, and in ordinary conversation we might have x ‘better than’ y, x ‘more beautiful than’ y, x ‘like’ y, x ‘a brother of’ y, and so on. These are all instances of a sentence of the form ‘xRy’— x ‘is in the relation R to’ y; but, while we may know what the above specific instances of an R mean, what do we mean in general by a relation R? On reflection we can give a definition as follows. We would in principle know all we need know about a relation R, if we knew all pairs x, y such that xRy. Thus, we use the same idea as in Definition 3.4.1 and make the following definition.