For example, I have a set of A = {1,2,3}. To express relations on the set A, we need the Cartesian product A x A.
For example, I want to express a relation that is reflective and I will write R = {(1,1), (2,2), (3,3)} But I often heard that there is also a term or phrase that is called the smallest relation. What in this case will be the smallest relation and how it differs from the not smallest relation one by the example of the property of reflexivity (if possible).
As you say, a relation is a set of ordered pairs taken from the Ccartesian product of a set with itself. We can partially order relations by inclusion. The smallest one will be the one that is a subset of all the others. Among reflexive relations, the one you give is the smallest because those three pairs must be in every reflexive relation. You can add any other pairs you wish without spoiling the fact that the relation is reflexive.
As another example, the empty relation is the smallest of all relations.