Basic Properties of Rough Sets and Rough Membership Function1

(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. Next we state the proposition (2) For every reflexive relational structure L holds idthe carrier of L ⊆ the internal relation of L. One can check that every reflexive relational structure which is non discrete is also non trivial and every relational structure which is reflexive and trivial is also discrete. We now state the proposition (3) For every set X and for every total reflexive binary relation R on X holds idX ⊆ R. Let us observe that every relational structure which is discrete is also diagonal and every relational structure which is non diagonal is also non discrete. Let us note that there exists a relational structure which is non diagonal and non empty. We now state three propositions: 1This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.