Relation
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.
Binary Relations
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.
n-ary Relations
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.