1 Chapter 2 Imprecise Categories, Approximations and Rough Sets #6 ChenKuang(Andy) Yang

2 Introduction Rough Sets Approximations of Set Properties of Approximations Approximations and Membership Relation

3 Introduction Fundamental concepts of the theory of knowledge – Classifications – Categories. Categories are subsets which can be derived using knowledge from a given knowledge base.

4 Some categories can be definable in one knowledge base but is undefinable in another knowledge base. If a category is not definable in a given knowledge base, can it be defined “approximately” in the given knowledge base?  The vague categories.

5 Rough Sets Let, and R be an equivalence relation. – X is R-definable if X is the union of some R-basic categories. – X is R-undefinable otherwise. R-definable sets will be also called R-exact sets R-undefinable sets will be also called R-inexact or R- rough. Set is said to be rough in K, if X is R-rough for any. Rough sets can be defined approximately: upper and lower approximation.

6 Approximations of Set Some subsets cannot be expressed exactly by employing available knowledge  Approximation of a set by other sets. R-lower and R-upper approximation of X –

7 The lower and upper approximations can be also presented in an equivalent form – Is the set of all elements of U which can be certainly classified as elements of X; is the set of elements of U which can be possibly classified as elements of X.

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9 Proposition – X is R-definable if and only if – X is rough with respect to R if and only if

10 Properties of Approximations

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12 Example Given a knowledge base K=(U,R), where U={x 1,x 2,…,x 8 }, and an equivalence relation with the following equivalence classes:

13 i.e. Let X 1 ={x 1,x 4,x 7 } and X 2 ={x 2,x 8 } – =

14 Approximations and Membership Relation Definition of a set is associated with knowledge about the set, thus a membership relation must be related to the knowledge. – Lower membership relation – Upper membership relation Both membership relations are referring to our knowledge and is not absolute.

15 Properties of Membership Relations

16 Q&A