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More on Cartesian products

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Most of the set theory relations we will deal with within the context of philosophy will be binary relations. However, one can relate as one sets to as many other sets as one wishes. Keeping in mind that a relation is a subset of a Cartesian product (e.g., S X T), one can write, for example, S X T X B X A, etc. ad infinitum. Instead of a binary relation, for example, one could create a ternary relations.

Such a relation would be a subset of the Cartesian product of three sets. One could also create a relation that is a subset of the Cartesian product of four sets. This would be a quaternary relation. The number of sets in the Cartesian product determines what we refer to as the "arity" of the relation. This simply refers to the number of the places involved in the relation. "An n-ary relation is also referred to as an n-place relation. The arity of a relation is the number of its places. So the arity of an n-ary relation is n. Note that the arity of ar elation is sometimes referred to as its degree"(Steinhart, p. 26).

Steinhart, Eric. "More Precisely: The Mathematics You Need To Do Philosophy." Broadview Press, 2009.