Some Remarks on the Concept of Approximations from the View of Knowledge Engineering

Rs-Approximations In (Infinite) Universe

The approximation theory of RS is well known. For preciseness, we will recall the notion here. Let U be a classical set, called the universe. Let be a partition, namely, a family of subsets, called equivalence classes, that are mutually disjoint and their union is the whole universe U. Then the pair (U, ) is called approximation space in RS. Pawlak introduced following two definitions. Observe that Pawlak focus on finite universe. However we allow U to be infinite.

Let X be an arbitrary subset of the universe U.

Definition (RS) 1 Let E be an arbitrary equivalence class of R.

• 1.

  Upper approximation:

• 2.

  Lower approximation:

This definition is the formal form of the intuitive upper and lower approximations

Definition (RS)2 Let p be an arbitrary element of U.

• 1.

  Closure = {, if , then }; note that C[X] is a closed set in the sense of topological spaces.

• 2.

  InteriorI[X] = { such that & }.

In RS community, the previous definitions are directed generalized to Covering Cov by interpreting E as member of Cov .

Counter Intuitive Phenomena

In this section, we present some Counter Intuitive phenomena of approximations. The first example was generated to answer some questions raised in a conversation with Tian Yang, Guangming Lang, Jing Hao from Hunan University.

• Example 1. Let the universe U be the real line. Let us consider the collection COV of all open half lines, namely, the sets of the following form {u | u < a} and {u | a < u} for a U. These half lines form a sub-base of the usual topology in real line; Here the “usual topology “ is a technical term referring to the topology of commonly known closure of the whole set.