Elements of Set Theory: n-ARY relations

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Elements of Set Theory: n-ARY relations

You might be wondering why do relations only have to do with ordered pairs. We can actually extend the ideas of ordered pairs to the case of ordered triples, and, more generally, to ordered n-tuples.

We can still view triple as ordered pairs if defined as follows,

Similarly, for ordered quadruples we can define it as,

By logical reasoning, we could continue in this way to define ordered quintuples or ordered n-tuples for any particular alt For uniformity is is convenient to define also the 1-tuple alt

Now, we can define an n-ary relation on A to be a set of ordered n-tuples with all components in A.

Thus a binary relation (2-ary relation) on A is just a subset of alt . And a ternary (3-ary) relation on A is just a subset of alt .

Note, however, that for alt then any n-ary relation is actually a relation on A. A special case occur for n=1, for the unary relation on A, then we just have a subset of A.

^{Disclaimer: this is a summary of section 3.3 from the book "Elements of Set Theory" by Herbert B. Enderton, the content apart from rephrasing is identical, most of the equations are from the book and the same examples are treated. All of the equation images were screenshots from generated latex form using typora}