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according to wikipedia the definition of relation is a set ordered pairs that is subset to cartesian product.

My question is ''Is this all about relations ?'' so it's just ordered pairs even if it doesn't carry any type of notion or relation between them.

So if there a set of pairs between unrelated sets then these pairs define a relation even thought there is not an relation between them according to our intuition like an example:

R={(cat,12),(T,e),(dog,0)}

So does the mathematical meaning of relation differs completly of our intuition about relations?

  • @JoséCarlosSantos So a relation can be any set of ordered pairs even thought the ordered pairs doesn't carry a definite relation ? –  Apr 24 '22 at 10:04
  • Yes, that is correct. – José Carlos Santos Apr 24 '22 at 10:05
  • @JoséCarlosSantos R={(cat,12),(T,e),(dog,0)}, so this is a well defined relation,Am I correct ? –  Apr 24 '22 at 10:08
  • It is a well-defined relation on any set which contains the set ${\text{cat},T,\text{dog},12,e,0}$. – José Carlos Santos Apr 24 '22 at 10:11
  • Note that the interpretation is that “cat” is related to $12$, $T$ is related to $e$ and “dog” is related to $0$. Yes, a relation is just a subset of a Cartesian product, but it’s a shift in perspective when we call it a relation. – Milten Apr 24 '22 at 10:25
  • @JoséCarlosSantos just another one question I want to get sure that I'm thinking correctly, one of the famous relations is (<) and we defined the relation as sets of ordered pairs so the (<) Relation is a sets of ordered pairs and not the idea of the ''less-than'' itself so the correct thing to say it's an ordered pairs not the ''[is-less-than] property itself'' Am I correct ? –  Apr 24 '22 at 13:03
  • Yes. For instance, in the set ${0,1,2,3}$ the $<$ relation is the set$${(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)}.$$ – José Carlos Santos Apr 24 '22 at 13:10

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Actually, any subset of a cartesian product does in fact define a relation. So, assuming the elements of your $R$ came from the same cartesian product set, yes they are a relation. The relation just isn't obvious. If they're from different sets, then by definition they don't form a relation, as they're not a subset of a cartesian product. That in mind, we could still form a relation if we took the union of those sets.

  • So they can be any set of ordered pairs even if the pairs don't have a definite relation between them but they still define a relation ,Am I correct ? –  Apr 24 '22 at 10:02
  • The point is that once you have specified the set of pairs, you have DEFINED a "definite relation between them"! What else do you want? – George Ivey Apr 24 '22 at 11:50
  • @GeorgeIvey What I mean by a definite relation is something like (=) or (<) something that we know intuitively we all know the idea of some number is less than the other but my question was if there is an ordered pair carries no intuitive idea between it's pairs does this also we can call it a relation something like this R={(cat,12),(T,e),(dog,0)} there is no intuitive idea between it's pairs –  Apr 24 '22 at 12:49
  • So this doesn't actually have anything to do with mathematics? – George Ivey Apr 25 '22 at 18:14