On wikipedia's 'Arity' page I noticed the statement,
The arity of a relation (or predicate) is the dimension of the domain in the corresponding Cartesian product.
I do not understand what this means and do not see an explanation on that page.
What is a relation's corresponding Cartesian product?
What is the dimension of this Cartesian product's domain?
It is the number of inputs to the relation or predicate. The term is also used with functions, while I have not seen relations with other than two inputs. The relation "less than" needs two values to make a complete sentence such as $x \lt y$, so has arity two. The function $\sin$ takes one input to make a value like $\sin x$ so has arity one. The distance function in $\Bbb R^3, \sqrt{x^2+y^2+z^2}$ takes three inputs, so has arity three.
By definition:
an $n$-ary relation on $n$ sets, is any subset of Cartesian product of the $n$ sets.
so, as an example: $$ R_1(x,y,z)=\{(x,y,z)\in \mathbb{R}^3 : x^2+y^2+z^2\ge 1\} $$
is a relation of arity=$3$
