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For a fixed x, I(x) will be used to denote a set of y’s such that I(x,y) holds. In other words I(x) = [x]I is an equivalence class induced by the relation I

For any set X in X we assign two sets called the I-lower and I-upper approximation of X:

lower approximation

upper approximation

The resulting approximations considered together are called a rough set X with respect to I. More precisely, we will write this down as

where denotes a rough set. The difference set ∂ taken as a difference between I∗(X) and I∗(X),

plays a role of a boundary region of . Its conveys the genuine notion of roughness. If ∂ = Ø then is exact (with respect to I).

The underlying philosophy of vagueness as it is captured by rough sets comes from Frege:

… the concept must have a sharp boundary. To the concept without a sharp boundary there would correspond an area that had not a sharp boundary-line all around…

There is an important point to be raised: rough sets are dwelled upon the indiscernibility relation. By changing the indiscernibility relation one can have a number of different rough sets implied by the same X - all of them are indexed by I.

The following is a simple example of a rough set. Consider a real line (real numbers) R and define the indiscerninbility relation of being approximated by the same integer number

I(x, y) = {x and y are approximated by the same integer number}

The equivalence class [x]I comprises all y’s such that they are approximated by the same integer as x. This gives rise to the intervals (A-1, A0, A1, etc.) shown in Fig. 3.9. We can regard I as a model of a sensor of a finite resolution.


Figure 3.9  X (i) and its rough set representation - lower and upper bound (ii)

Consider, for instance, X represented as a set (interval) [0.2, 1.6], see again Fig. 3.9. The lower bound of comes in the form of A1, I*(x)= {A1}. The upper bound includes A0, A1, and A2, I∗(x) = {A0, A1, A2}.

The membership function of a rough set is defined as

It assumes values between 0 and 1. Depending on a location of x, we distinguish several situations

Equivalently, we can describe the bounds of via the expressions

The membership approach focuses on the numerical aspects of rough sets while the characterization provided by their bounds reveals an underlying topological structure. It should be noted that the definition of the membership function of is secondary to the notion of the rough set itself - one could come up with another definition of the membership functions.

3.14. Rough sets and fuzzy sets

One can establish several intriguing and useful links between rough sets and fuzzy sets. We elaborate on the two of them in more detail

processing of fuzzy sets in the setting of the indiscernibility relation. Assuming that X is a fuzzy set, it is processed in the context of the assumed indiscernibility relation. The lower and upper bounds need to be slightly reformulated to capture the feature of intermediate membership values of X. This leads us to the possibility and necessity measures

hence X∗ is a fuzzy set whose membership function is defined via the corresponding possibility values,

Similarly, the membership function of X∗ (which now becomes a fuzzy set) is computed by determining the necessity measure

The resulting construct captures both the notion of roughness (originating from the indiscernibility relation) and fuzziness - this element arises because of the fuzzy set X processed in the context characteristic for rough sets.

fuzzy sets as elements of the indicernibility relation The previous indicernibility relation becomes then a similarity (as opposed to equivalence) relation. If X is just a set, it gives rise to a rough set with the fuzzy boundaries - their appearance is due to the fuzzy sets used in the similarity relation. If both X and the elements are implied by the indiscernibility (similarity) relation, then one can replace these fuzzy sets by their appropriate set approximations -induced Boolean partition (one of the possibilities is described in Section 3.16).


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