Basic Properties of Rough Sets and Rough Membership Function

Let A be a set. One can verify that 〈A, idA〉 is discrete. The following proposition is true (1) For every set X such that ∇X ⊆ idX holds X is trivial. Let A be a relational structure. We say that A is diagonal if and only if: (Def. 1) The internal relation of A ⊆ idthe carrier of A. Let A be a non trivial set. Observe that 〈A,∇A〉 is non diagonal. We now state the proposition (2) For every reflexive relational structure L holds idthe carrier of L ⊆ the internal relation of L.