The notion of relations (also known as predicates) is one of the most important fundamental concepts of set theory. The most common of these are the binary relations, so we begin with them. Once they have been established, we generalize to -ary relations, which we apparently don't come across often, but which occur implicitly very frequently in mathematics.
A binary relation between a space and a space is formally defined as a subset of . If and , we say is related to under , and write , or, more commonly, , iff .
For a more detailed treatment, see Binary relation.
An -ary relation over the sets is a subset of . If for , we say are related under , and write (unfortunately though, the other short hand breaks down here) iff . If , we say is an n-ary relation over .
A very common example of an -ary relation is a linear constraint over a vector space for some field : , where is an element of the vector space and are scalars.