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it seems that everything I read about relations stresses that they are a subset of a cartesian product.
Sometimes, they will say that a cartesian product itself is a relation. This seems confusing to me. If a cartesian product can be considered to be a relation, than we should say that a relation is a cartesian product or a subset thereof.
If, OTOH, there is a reason why it has to be a subset, but not the product itself, than we should have an explanation as to why.
Also, is there any rule as to how this subset supposed to be formed? TIA.

Mordechai
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  • Read this Wikipedia post about relations. Relations are defined as subsets of Cartesian product of two sets. Any subset of the Cartesian product A x B is a relation from A to B. The null set and the Cartesian product set itself, both are also subsets of the Cartesian product and thus are relations too. – insipidintegrator Sep 05 '22 at 06:06
  • besides, the insult of sending someone to a wikipedia, the comment is pointless as it ignores the question. That's the whole point. It would have been more precise to say a product and all of its subsets. Also, this still leaves the second part of the question about if there are any rules for deriving said subsets or is it completely arbitrary. – Mordechai Sep 05 '22 at 06:13
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    Every set is a subset of itself (but not a proper subset). – Karl Sep 05 '22 at 06:25

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A binary relation on sets $X,Y$ is a subset of the Cartesian product $X\times Y$. This is the definition of a relation; there are no other requirements. In other words, if you have a set $R$ with the property that every element of $R$ is a pair $(x,y)$ with $x\in X$ and $y\in Y$, then $R$ is a relation. A typical way to specify a relation is set-builder notation.

Karl
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  • now that I think about it, I remember seeing something about a relation having a meaning of a relationship between the domain sets. E.g. "less then" relation over X, Y would contain all ordered pairs x,y where x<y. So, it's not completely arbitrary. – Mordechai Sep 05 '22 at 13:07
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    It's arbitrary in that every subset of the Cartesian product can be seen as describing a relationship in that way. A relationship is nothing but a set of ordered pairs. – Karl Sep 05 '22 at 17:16