The first I refers to a given set, such as a real number, sometimes a complete set, just a code name.

The second possibility is terrible, give a joke:

2.3.3 Lower Approximation, Upper Approximation and Boundary Region of Sets

Rough set theory extends the classical set theory and embeds the knowledge used for classification into the set as a part of the set composition. Whether an object A belongs to set X depends on the existing knowledge, which can be divided into three situations: (1) Object A definitely belongs to set X; (2) Object A definitely does not belong to set X; (3) Object A may or may not belong to set X, and the division of sets closely depends on our understanding of the universe, which is relative rather than absolute. Given that a finite nonempty set u is a universe and I is a family of equivalence relations in u, that is, knowledge about u, then this binary pair K = (U, I) is called an approximate space. Let X be an object in U, a subset of U, and I (x) represents the set of all objects that are indistinguishable from X, in other words, it is determined by X.

The equivalence class, that is, every object in I (x) has the same properties as X. 。

The lower approximation of the set x relative to I is defined as:

I* (X) = {x ∈U: I (x) I *(X) In fact, the largest set of objects belonging to X is determined according to existing knowledge, sometimes called.

Is the possible area of x, and is recorded as PO S (X). Similarly, according to the existing knowledge, it is definitely not X.

The set of objects is called the negative region of x, and is denoted as N EG (X).

The upper approximation of the set x with respect to I is defined as

I3 (X ) = {x ∈U : I (x ) ∩ X ≠ 5 } (2)

I3 (X) is the union of all nonempty equivalence classes I (x) that intersect with X, and it is the smallest combination of those objects that may belong to X..

Settings. Obviously, i3 (x)+n eg (x) = Universe u.

The boundary region of set x is defined as

BND (X ) = I

Article 3, paragraph 10, paragraph 3, paragraph 10, item 3

BND (X) is the difference between the upper approximation and the lower approximation of the set X. If BND (X) is an empty set, it is said that X is clear to I.

(crisp); On the other hand, if BND (X) is not an empty set, then set X is called rough set about I. 。

Concepts such as lower approximation, upper approximation and boundary region are called resolvable regions, while boundaries contain.

Approximate characteristics of vague sets. Roughness can be calculated by the following formula.

A 1

=

Article 3, paragraph 10

I

Article 3, paragraph 10, item 4

Where # represents the cardinality of the set #, and for a finite set, represents the number of elements contained in the set.

Obviously 0≤A

1 (X) ≤ 1, if a

1 (X) = 1, then the set x is clear relative to I, if a.

1 (X ) 0} (7)

BND(X)= { X∈U:0 & lt； Liquid ion exchange

(x)& lt； 1} (8)

From the above definition, we can see the two concepts of "vague" and "Uncertain ty" in rough set theory.

The relationship between concepts: "fuzzy" is used to describe a set, indicating that the boundary of the set is unclear; "Uncertainty" describes.

Element means that whether an element belongs to a set is uncertain.

This involves rough sets, right? ...