A THEORY OF FUZZY SETS BASED ON CUT SET WITH PARAMETERS

In this paper, we first discuss the relationships among the existing four cut sets, and introduce the unification of these four cut sets as one cut set with parameters. Then, we give out the properties of the cut set with parameters, and establish the appropriate decomposition theorem and representation theorem of fuzzy sets.


Introduction
Since the concept of fuzzy set was first introduced by Zadeh (see [1]), many discussions about fuzzy set are proposed.Based on the definitions of cut set and strong cut set of fuzzy set (see [2]), the cut sets, decomposition theorem, and representation theorem are all play very important roles in fuzzy reasoning (see [3] and [4]), fuzzy algebra (see [5] and [6]), fuzzy topology (see [7]), and fuzzy measure, see [8] [9] and [10].In paper [11], the authors introduced three new cut sets of the fuzzy sets and had obtained four new decomposition theorems and four new representation theorems.And then, in paper [12], four kinds of cut sets and eight mappings on intuitionistic fuzzy sets are introduced, based on which four decomposition theorems and four representation theorems on intuitionistic fuzzy sets are obtained.
Based on paper [12], the equivalence of the four decomposition theorems based on different cut sets for intuitionistic fuzzy sets is proved in the paper [13], as well as the equivalence of the four representation theorems.In fact, the equivalence of the four decomposition theorems and that of four new representation theorems based on different cut sets for fuzzy sets in paper [11] can be proved similarly.
This paper contains three major parts.It first discusses the relationships among the four cut sets in paper [11], then introduces three new cut products and gives the relationships among the cut products in Section 2. In section 3, we introduce the unification of these four cut sets as one cut set with parameters, and give out the properties of the cut set with parameters.Then, in section 4 and section 5, by using the cut set with parameters, we establish the appropriate decomposition theorem and representation theorem of fuzzy sets.

Preliminary
Let X be a set, and A is a mapping on x → A(x), we call that A is a fuzzy set on X, see [1].In this paper, the class of all fuzzy sets on X is denoted as F(X).P(X) denotes the power set of X, λ ∈ I = [0, 1].The following are the definitions of four cut setsA λ ,A λ ,A [λ] ,A [λ] and four strong cut setsA λ , A λ , A [λ] , A [λ] in paper [2] and [11]: Definition 1.1.LetA ∈ F(X), λ ∈ I, then: (1) A λ = {x|x ∈ X, A(x) ≥ λ}, and A λ = {x|x ∈ X, A(x) > λ} are called λ-upper cut set and λ-strong upper cut set of fuzzy set A respectively.

Relationships Among Four Cut Sets and that of Four Cut Products
Let X be a set, A ∈ F(X), λ ∈ I = [0, 1], and λ c = 1 − λ.By the Definition 1.1, it is not difficult to prove the relationships among these cut sets as follows: The definition of cut product λA we have known is that λA ∈ F(X), and (λA)(x) = λ ∧ A(x), where A ∈ F(X), λ ∈ I [2] .In particularly, if , it is the fuzzy subset λB defined in [11].Similarly, thinking about the other three fuzzy subsets λ [11], where (λ , we can define three new cut products: Definition 2.1.LetA ∈ F(X), λ ∈ I, we define: We call the λ • A, λ • A and λ♦A cut products exactly as λA.Then, the following property is clear.
Proof.We only prove λA = (λ c • A) c , and the others can be proved by the same method.For any x ∈ X,λ ∈ I, we have( By the Property 2.1 and 2.2, the equivalence of the four decomposition theorems and that of four new representation theorems based on different cut sets for fuzzy sets in paper [11] can be easily proved using the similar method as in paper [13].

A Cut Set with Parameters
Considering the relationships among the four cut sets provided in Property 2.1, we find the four cut sets can be unified as one cut set with parameters as following. Moreover, is the λ-strong cut set with parameters of fuzzy set A. Since Then, when α = β < 1 2 , In particular, when α = β = 0, So, the λ-cut set with parameters is a promotion form of the four cut sets in paper [11].That is also can be said that the λ-cut set with parameters is a unification form of the four cut sets in paper [11].
(3) Let the mapping H : Here t∈Φ It is easy to prove that the four decomposition theorems in paper [11] are particular cases of the above two theorems.

Conclusions
In this paper, a cut set with parameters is presented as the unification of the existing four cut sets and its properties are discussed.Based on the cut set with parameters, the appropriate decomposition theorem and representation theorem of fuzzy sets are established.These discussions extended the theories of fuzzy sets.