On the Normed Space of Equivalence Classes of Fuzzy Numbers

We study the norm induced by the supremum metric on the space of fuzzy numbers. And then we propose a method for constructing a norm on the quotient space of fuzzy numbers. This norm is very natural and works well with the induced metric on the quotient space.

In the classical mathematics, if is a normed space with norm ‖ ⋅ ‖, it is readily checked that the formula ( , ) = ‖ − ‖, for , ∈ , defines a metric on . Thus a normed space is naturally a metric space and all metric space concepts are meaningful. However, we will show that such proposition does not hold true for the well known supremum metric on the space of fuzzy numbers. To overcome this weakness, we will consider the quotient space of fuzzy numbers up to an equivalence relation which is introduced by Mareš [20,21] and is studied extensively by many researchers [4,12,[22][23][24]. We will propose a method for constructing a norm on the quotient space of fuzzy numbers. This norm is very natural and works well with the induced metric on the quotient space.

Preliminaries
for each ∈ R. For any , ] ∈ F and ∈ R, owing to Zadeh's extension principle [7][8][9], scalar multiplication and addition are defined for any ∈ R by For any ∈ F, we define the fuzzy number − ∈ F by − = (−1) × , that is, − ( ) = (− ), for all ∈ R. Since the addition does not possess an inverse subtraction, F is not a real vector space. We say that a fuzzy number ∈ F is symmetric [20], if for all ∈ R, that is, = − . The set of all symmetric fuzzy numbers will be denoted by S.

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The Scientific World Journal Definition 1 (see [22]). Let , ] ∈ F. We say that is equivalent to ] and write ∼ ] if and only if there exist symmetric fuzzy numbers 1 , 2 ∈ S such that The equivalence relation defined above is reflexive, symmetric, and transitive [20]. Let ⟨ ⟩ denote the equivalence class containing the element and denote the set of equivalence classes by F/S. By the level set representations for fuzzy numbers, one can easily prove the following lemmas.

Main Results
In this section, we will give a norm structure on F/S which is compatible with a metric. For the family of fuzzy numbers F, the ∞ -metric is induced by the Hausdorff metric as [17] (ii) For all ∈ F and ∈ R, we have (iii) For all , ] ∈ F, we have that We conclude that ‖ ⋅ ‖ is a norm on F.
Although ‖ ⋅ ‖ is a norm on F, the function : (iii) In order to prove the triangle inequality, for any fixed ∈ [0, 1], and for any , ], ∈ F, we only proof the following six cases. Similarly, the others can be proved.
The Scientific World Journal Consequently, we have that From what is proved above, we can get that (iv) Since ∉ R, there exists 0 ∈ [0, 1] such that ( 0 )− ( 0 ) > 0. Thus we have that In order to induce a metric which is compatible with the norm, we consider the quotient space of fuzzy numbers. It is very natural to define a function ‖ ⋅ ‖ : F/S → R as for each ⟨ ⟩ ∈ F/S.

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The Scientific World Journal (ii) For all ⟨ ⟩ ∈ F/S and ∈ R, we have We conclude that ‖ ⋅ ‖ is a norm on F/S.
We conclude that is a metric on F/S.

Conclusions
In this present investigation, we studied the norm induced by the supremum metric ∞ on the space of fuzzy numbers. And then we proposed a method for constructing a norm on the quotient space of fuzzy numbers. This norm is very natural and works well with the induced metric on F/S. The works in this paper enable us to study the fuzzy numbers in the new environment. We hope that our results in this paper may lead to significant, new, and innovative results in other related fields.