- Reflexive Relation

- Symmetric Relation

- Antisymmetric Relation

- Transitive Relation

I know how to find all these relations when a single set is being considered.

However, what if the sets are different? For example, consider the following two sets of same size,

A = {1, 2}

B = {2, 3}

or A = {1, 2}

B = {3, 4} (totally different elements from first set)

Then their possible relations will be all the subsets of their Cartisean Product.

Can anyone of those relations be reflexive, symmetric, antisymmetric, transitive? If I check from Matrix Representation of those relations, it shows that those relations do have all these properties. However, from definition, it tells you that not a single relation has any of these properties... Can anyone verify this please?

Also, if they do have all these properties, then what if we use sets of different sizes then? Will they also satisfy all these properties?

Thank you in advance.