- A binary relation is a set of ordered pairs of the elements of some other set.
That is the first definition I learned for binary relations.
Slightly more generally, a binary relation is a subset of a cartesian product of two sets, and In symbols, Of course and could be the same, but that’s not always the case.
I have long used the adjectives, 2-place, binary, and dyadic pretty much interchangeably in application to relations but I developed a bias toward dyadic on account of computational contexts where binary is reserved for binary numerals.
Once again, partly due to computational exigencies, I would now regard this first definition as the weak typing version.
The strong typing definition of a -place relation includes the cartesian product as an essential part of its specification. This serves to harmonize the definition of a -place relation with the use of mathematical category theory in computer science.
When I get more time, I’ll go through the material I linked on relation theory in a slightly more leisurely manner …