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.