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I read the following on the wikipedia article on binary relations:

A binary relation R between arbitrary sets (or classes) X (the set of departure) and Y (the set of destination or codomain) is specified by its graph G, which is a subset of the Cartesian product X × Y.

Is it really necessary to define relations in terms of graphs? Why do we need to introduce this aditional concept at this point? It seems to me that graphs are distinct mathematical objects with a lot more baggage than a mere subset which may be all we need at this point. What's wrong with simply saying:

A binary relation R between arbitrary sets (or classes) X (the set of departure) and Y (the set of destination or codomain) is a subset of the Cartesian product X × Y.

Marcus Junius Brutus
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    If you want to think of functions as a special case of relations, it's important to think of functions as distinct from (but recoverable from) their graphs. Similarly, if you want to think of relations as generalized functions, you may (or may not) want to define them as "nondeterministic partial functions": functions that return as output a subset, possibly empty, of the codomain. The relationship between this "function" and its graph is given by currying. – Qiaochu Yuan Sep 17 '17 at 04:17
  • @QiaochuYuan your comments appears to be very insightful (and your reputation also suggests that) but it's too deep for me. – Marcus Junius Brutus Sep 17 '17 at 15:11
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    By definition, a binary relation between A and B is a subset of AxB. What is a graph? A table of departures and arrivals? That definition appears in need of editing. – William Elliot Sep 18 '17 at 07:39
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    The word "graph" is causing understandable confusion here. The quoted article is using "graph" simply to mean the set of ordered pairs of the relation. The article is poorly worded but intends to say what the questioner suggests it should. – John Stell Feb 22 '23 at 08:28

1 Answers1

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For any relation $R$, the following is true for some set $U$:

$$R\subseteq U^2$$

Then you can take the domain and range of the relation by defining two sets with certain properties based on $R$.

Garmekain
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