It is convenient to begin with the definition of a -place relation, where is a positive integer.
Definition. A -place relation over the nonempty sets is
a -tuple where is a subset of the cartesian product
Several items of terminology are useful in discussing relations.
- The sets are called the domains of the relation with being the domain.
- If all the are the same set then is more simply described as a
-place relation over
- The set is called the graph of the relation on analogy with the graph of a function.
- If the sequence of sets is constant throughout a given discussion or is otherwise determinate in context then the relation is determined by its graph making it acceptable to denote the relation by referring to its graph.
- Other synonyms for the adjective -place are -adic and -ary, all of which leads to the integer being called the dimension, adicity, or arity of the relation