## Relations & Their Relatives • Discussion 21

AS:
Let me underline an important point:  first of all, we have found in nature and society one or another relation and ask how many members each example of this relation can have?  i.e. arity is a feature of relation itself.  So […] we come here to the logic of relations and its discovery.  For me, examples of relations of different arity from one or another domain would be great.

Here’s a first introduction to $k$-adic or $k$-ary relations from a mathematical perspective.

Here’s a few additional resources and assorted discussions with folks around the web.

More than anything else it is critical to understand the differences among the following things.

1. The relation itself, which is a mathematical object,
a subset embedded in a cartesian product of several
sets called the “domains” of the relation.
2. The individual $k$-tuple, sometimes called an “elementary relation”,
a single element of the relation and therefore of the cartesian product.
3. The syntactic forms, lexical or graphical or whatever,
used to describe elements and subsets of the relation.
4. The real phenomena and real situations, empirical or quasi-empirical,
which we use mathematical objects such as numbers, sets, functions,
graphs, groups, algebras, manifolds, relations, etc. to model, at least
approximately and well enough to cope with the realities in practice.

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