Rough Sets:
We first introduce the concept of the lower approximation in rough set theory. Subsequently, we present the notion of the upper approximation. Thereafter, to improve understanding, we provide a representative rough set example. Through this illustrative example, we clearly demonstrate the process of data granulation and further explain how data reduction is achieved.
What is a Rough Set?
A Rough Set is used to represent a set of objects that cannot be precisely defined using the available information. It approximates a “target set” by using two precise sets: the Lower Approximation and the Upper Approximation.
• Indiscernibility Relation: When two objects are identical based on their properties, they are indiscernible.
• Elementary Sets: Groups of objects that look exactly the same.
The Two Approximations
- Lower Approximation (RX): The set of objects that certainly belong to the target set.
- Upper Approximation (R̄X): The set of objects that possibly belong to the target set.
- Boundary Region: The difference between the two. If the boundary is empty, the set is crisp; otherwise, it is “Rough.”
Clear Example: Patient Diagnosis
Suppose we have 5 patients {p1, p2, p3, p4, p5}. Based on their symptoms, we group them into elementary sets (objects that look the same):
Elementary Set 2: {p3} (Unique symptoms)
Elementary Set 3: {p4, p5} (Identical symptoms)
Now, let the target set X be patients who actually have “Flu”: X = {p1, p2, p3, p4}.
{p1, p2} is in X. {p3} is in X.
RX = {p1, p2, p3}
2. Upper Approximation (R̄X): Which groups have at least one member in X?
{p1, p2} has members in X. {p3} has members in X. {p4, p5} has p4 in X.
R̄X = {p1, p2, p3, p4, p5}
3. Boundary Region: R̄X minus RX = {p4, p5}. (Uncertain region).
Operations of Rough Sets
Operations on Rough Sets are performed by calculating the approximations of the resulting set.
| Operation | Lower Approximation Rule | Upper Approximation Rule |
|---|---|---|
| Union (A ∪ B) | R(A) ∪ R(B) | R̄(A ∪ B) |
| Intersection (A ∩ B) | R(A ∩ B) | R̄(A) ∩ R̄(B) |
The Intersection Lower Approximation is the common “certain” elements: {p1}.
Rough Numbers (RN)
A Rough Number is defined by its lower limit and upper limit, derived from the approximations. It is written as:
Operations of Rough Numbers
Let RN1 = [L1, U1] and RN2 = [L2, U2].
Example: [2, 5] + [3, 4] = [5, 9]
Example: [10, 15] – [2, 5] = [10-5, 15-2] = [5, 13]
Example: [2, 3] × [4, 6] = [8, 18]
Designed by: Dr. M.U. Mirza
Mathematical Researcher & Educator