Q2. What is the motivation behind the theory of rough sets?
The theory of rough sets is motivated by practical needs in classification, concept formation, and data analysis with insufficient and incomplete information [12–15].
Q3. How do the authors define operations on binary relations?
The authors define operations on binary relations through set-theoretic operations:∼R = {(x, y) | not xRy},R ∩ Q = {(x, y) | xRy and xQy},R ∪ Q = {(x, y) | xRy or xQy}.
Q4. What is the inverse serial condition for (P1)?
A set of necessary and sufficient condition for (P) is:(P1) n is inverse serial,(P2) for all x, y ∈ U, either n(x) = n(y) or n(x) ∩ n(y) = ∅.Condition (P1) is equivalent to saying that ⋃x∈U n(x) = U .
Q5. What generalization is used to decide the memberships of a subset X?
For this generalization, only the neighborhood of x is used to decide the memberships of x in the lower and upper approximation of a subset X.
Q6. What are the properties of the neighborhood operators?
According to property (U2) and equations (15) and (16), upper approximation operators can be interpreted using neighborhood operators:aprRp(X) = Rs(X), aprRs(X) = Rp(X), aprRp∧s(X) = Rp∧s(X), aprRp∨s(X) = Rp∨s(X).
Q7. What are the different relations between R and its inverse?
For a relation R and its inverse R−1, the application of operators ∼, ∩, and ∪ produces 16 different relations, such as ∼R, ∼R−1, and R ∪ ∼R−1.