At the beginning of Sipser's Theory of Computation we get the following two definitions:
A predicate or property is a function whose range is $\{TRUE, FALSE\}$. For example, let even be a property that is $TRUE$ if its input is an even number and $FALSE$ if its input is an odd number. Thus $even(4) = TRUE$ and $even(5) = FALSE$.
A property whose domain is a set of k-tuples $A \times \cdots \times A$ is called a relation, a k-ary relation, or a k-ary relation on $A$.
My question: isn't Sipser's definition of relation nonstandard here (I am used to the definition of a relation as any subset of of a Cartesian product of sets)? That is, he defines a relation as a very specific type of function (a function with a specific domain as a Cartesian product of a given set with itself $n$ times and a range of $\{T,F\}$). Is this a common definition in computability theory?