General Letters in Mathematics (GLM)

In this paper, we introduce F( R p ) , the collection of all ﬁnite sets of fuzzy subsets in F bc ( R p ) , which is the class of the upper semicontinuous fuzzy subsets of R p with bounded convex compact closure of the supports. Then we deﬁne set-fuzzy-set multifunctions and discuss various kinds of them such as monotone, fuzzy measure, subadditive, null-additive, null-null-additive, multisubmeasure and null-continuous. We introduce the set-fuzzy-set-norm variation of a set-fuzzy-set multifunction and present some properties between a set-fuzzy-set multifunction and its variations such as fuzzyness, continuity from below, null-additivity, and null-null-additivity.

As a recent research, Akram [1] introduced the notion of m-polar fuzzy matroids and discussed certain applications of m-polar fuzzy matroids in decision support systems, ordering of machines and network analysis. He presented in [2] certain metrics in m-polar fuzzy graphs including antipodal m-polar fuzzy graphs and self median m-polar fuzzy graph of given m-polar fuzzy graphs. Sarwar and Akram [35] introduced the concept of bipolar fuzzy matroids and applied it to graph theory and linear algebra. They also described certain applications of bipolar fuzzy matroids in decision support system and network analysis.
While all of the researches about the set multifunctions focused on ordinary sets, in this paper, we demonstrate the possibility of improving current definitions using the finite sets of fuzzy sets and introduce a set-fuzzy-set multifunction and its set-fuzzy-set-norm variations. This paper is organized as follows: in Section 2, we define fuzzy-set-norm on F(R p ), the family of all fuzzy subsete of R p . According to Due Gil et. al. [25], we consider the class of the upper semicontinuous fuzzy subsets of R p with bounded convex compact closure of the supports denoted by F bc (R p ). As an useful example, we introduce the fuzzy-set norm |.| a on F bc (R 2 ). In section 3, we deal with F(R p ) which denotes the collection of all finite sets of fuzzy subsets in F bc (R p ). We prove any fuzzy-set-norm on F bc (R p ) induces a set-fuzzy-set-norm on F(R p ). In Section 4, we consider a nontrivial fuzzy subset T of M and a ring C of fuzzy subsets less than or equal to T . We define various kinds of set-fuzzyset multifunction µ : C → (F(R p ), , | . |) such as monotone, fuzzy measure, subadditive, null-additive, null-null-additive, multisubmeasure and null-continuous. We prove four theorems about the relations between them and basic results. In section 5, we install the set-fuzzy-set-norm variation of a set-fuzzy-set multifunction and present some properties between a set-fuzzy-set multifunction and its variation such as fuzzyness, continuity from below, nulladditivity, and null-null-additivity.

Fuzzy-set-norms
Consider the p-dimensional Euclidean space R p and | . | as the usual associated norm. Zadeh [40] denoted by K(R p ), the class of the nonempty compact subsets of R p and by K bc (R p ), the subclass of bounded convex sets in K(R p ). The space K(R p ) can be represented by a semilinear structure using the Minkowski sum and the product by a scalar K We denote by F bc (R p ) Due Gil et. al. [25], the class of the upper semicontinuous fuzzy subsets of R p with bounded convex compact closure of the support.
The space F bc (R p ) can be endowed with a semilinear structure by means of the sum and the product by a scalar based on Zadeh's extension principle [40]. Applying this principle for elements in F bc (R p ) states that for E, F ∈ F(R p ) and α ∈ R.

Example 2.2.
If (R p , ∥ ∥) is a normed space, then the function is a fuzzy-set-norm on (F(R p ), ), called the supremum fuzzy-set-norm on F(R p ).
Since E(x, y) ∈ [0, 1] and suppE is a bounded convex compact subset of R 2 , this integral equals to a nonnegative real number. We show that | . | a satisfies three conditions of Def. 2.2: hence supp(αE) and supp(E + F) are equal to the bounded convex compact subsets suppE and suppE ∪ suppF respectively. By using linearity of integration, we see (ii) and (iii) hold. Also for every fuzzy Therefore, | . | a is a monotone fuzzy-set-norm on (F(R 2 ), ).

Set-fuzzy-set-norms
The largest set class of set of fuzzy set values, we will deal with in this paper, is F(R p ), which denotes the collection of all finite sets of fuzzy subsets We denote (F(R p ), ) since we consider the usual fuzzy set inclusion ⊆ as an order on F(R p ). The space F(R p ) can be endowed with a semilinear structure by the following defined sum and the product by a scalar: 1) Using (2.1), we define every α ∈ R and eachẼ ∈ F(R p ) in the form (3.1), 2) Using (2.2), we define eachẼ ∈ F(R p ) in the form (3.1) and each ,F in the form Without loss of generality, assume that k k ′ Proof. Since the finite sum of several non-negative real numbers is a non-negative real number, this norm is well defined. Also we show that it satisfies the conditions of Def. 2.2:

Set-fuzzy-set multifunctions
Let T be a nontrivial fuzzy subset of M, then any fuzzy subset of M, which is less than or equal to T , is called an fuzzy subset of T . Let C be a ring of fuzzy subsets of T and F(T ) be the family of all fuzzy subsets of T.
(iv) strongly fsn-continuous if lim n→∞ |µ(A n )| = 0 for every (A n ) n∈N * ⊂ C such that A n ⊇ A n+1 for all n ∈ N * and µ( Definition 4.4. C is a σ-ring if the following conditions are met: Proof. Let (A n ) n∈N * be a sequence of pairwise disjoint sets of C and let B n = ∪ ∞ k=n A k for all n ∈ N * . Then B n ∈ C for every n ∈ N * and B n ↘0. Since µ is fsn-continuous, then lim n→∞ |µ(B n )| = 0, thus lim n→∞ |µ(A n )| = 0. So µ is fsn-exhaustive. Theorem 4.6. Let C be a σ-ring and µ : C → (F(R p ), , | . |) be fuzzy measure. If µ is null-null-additive and strongly fsn-continuous, then µ is null-continuous.
We define a subsequence (A n k ) of (A n ) as follows. Let n 1 = 1. For every k ∈ N * since µ(A n k ) = {0} and A n k ∪ (A\A n ) ↘ A n k , when n → ∞, by the fact that µ is strongly fsn-continuous, we can choose n k+1 so that n k+1 > n k and of fuzzy subsets of T that satisfies three conditions: It is seen that .
We have We define the fuzzy set D, B : M → [0, 1], by: and the fuzzy sets A 1 , A 2 : M → [0, 1] by: b) It is seen that µ is not monotone: c) It is seen that II) If µ is a multisubmeasure of finite variation and µ is continuous from below on C thenμ is continuous from below.
Since B i ∩ A n ↗ B i and µ is continuous from below for every i ∈ {1, . . . , m}, there is n i (ε), n i∈N so that for every n > n i . Let n 0 = max 0 i m n i . Then for all n > n 0 we have Since B i is arbitrary, we obtainμ(A) < ε +μ(A n ) for every n > n 0 . This shows that Therefore,μ is continuous from below on C. II. Let (A n ) n∈N * ⊂ C so that, A n ↗ A. Sinceμ is superadditive, we have: Since µ is a multisubmeasure, we have: Let C = F(T ). We know everyone in Iran has a ten-digit number called national code. So we can assign a fuzzy function to each individual, which is actually a fuzzy point using the same code. For example assume that my national code is 0385234673. Then my fuzzy point will be Suppose the tax department wants to determine which apartment or house or land each person owns and how much his share is. Therefore, the function µ : C → (F(R 3 ), , | . | s ) assigns to each fuzzy point associated with each person's code, a finite set of fuzzy functions. When S = suppE i is a piece of a quadratic surface f(x, y, z) = c, then we define the fuzzy-set-norm | . | t on F bc (R 3 ) by: g(x, y, z)E(x, y, z) dσ. (5.5)