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I have problems understanding these properties.

Let us consider the set A whose Cartesian product is equal to A x A.

Let A = {1,2,3}.

Then:

The relationship with the reflexivity property should look like this: R = {(1,1), (2,2), (3,3)} - because for the set A x A, each element should be paired with itself and options such as R = {(3,3)} - it cannot be, so not every element of the set already corresponds to itself.

The relation with the symmetry property should look like this: R = {} or R = {(1,1)} or R = {(2,2)} or R = {(3,3)} or R = {(1,1 ), (2,2)} or R = {(1,1), (3,3)} or R = {(2,2), (3,3)} or R = {(1,1), (1,2), (2,1)} or R = {(1,1), (1,3), (3,1)} etc. - because there should be a symmetric pair (but I'm not sure if each element must be symmetric to itself from the set A x A, because in the property of reflexivity it should)

Now let's look at the sets A and B whose Cartesian product is A x B. Let A = {1,2,3}, and B = {3,4,5}.

Is it possible to build relationships with the property of reflexivity, or symmetry?

As for reflexivity, I believe that it is impossible, since not every element from the set A will correspond to an element from the set B (except for the pair (3,3))

As for symmetry, I’m completely confused: since I’m not very sure about the correct understanding of this property with A x A, then with A x B all the more so. But I will assume that this should be a symmetric relation: R = {(3,3)} or R = {} (which is the least by the symmetry property)

Help me to understand.

Andrés E. Caicedo
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MaximPro
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    "Now let's look at the sets A and B whose Cartesian product is A x B. Let A = {1,2,3}, and B = {3,4,5}. Is it possible to build relationships with the property of reflexivity, or symmetry?" -- By my understanding of relations, you would do them on a single set's product with itself. For example, $A \times A, B\times B, (A \times B) \times (A \times B)$ all work, but not two differing sets $A \times B$ by themselves. Not sure if it really matters in your case, but I figured it would be worth noting. – PrincessEev Oct 09 '19 at 05:30
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    These properties are defined for a relation on a single set, not a relation between two sets. Take a look at here. – Bumblebee Oct 09 '19 at 05:30
  • That aside, you seem to have a pretty decent grasp of everything involved unless I missed something. – PrincessEev Oct 09 '19 at 05:31
  • @EeveeTrainer yes this example that you made - of course it works because it resembles the following: AxB = C then CxC = R which, in meaning, is the same as AxA and BxB. Although I'm not interested in this. – MaximPro Oct 09 '19 at 05:44
  • I know, I was just stating them as other examples, to emphasize my point of the notion of a relation $R$ on a set $S$ as being a subset of the Cartesian product, i.e. $R \subseteq S \times S$. I don't know whether a term exists when the product is of two different sets (or, indeed, if that's interesting enough mathematically to be studied). EDIT: Though, I suppose the definition of a function applies, now that I think of it. $f : A \to B$ is analogous to $f \subseteq A \times B$ (where the set $f$ is just all the pairs $(x,f(x))$). But I don't recall functions being called reflexive/etc. – PrincessEev Oct 09 '19 at 05:46
  • Oh, I'm dumb, it would be a binary relation specifically. Ignore my ramblings, I'll have to look into this. – PrincessEev Oct 09 '19 at 05:51
  • @EeveeTrainer If I'm not mistaken, what you're talking about is called conformity. Relationships are a special case of conformity.I think the properties of relationships cannot be applied to conformity that are not relationships. Probably. I'm not sure that I give the correct terminology - English is not my native language. – MaximPro Oct 09 '19 at 05:54

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