A novel difference and derivative of linearly correlated fuzzy number-valued functions

3.1 A is non-symmetric

In essence, the operator Ψ _A is injective if and only if A ∈ ℝ F is non-symmetric. So Ψ _A is a bijection from ℝ 2 to ℝ F ( A ) . Therefore, the structure of the set ℝ F ( A ) depends on the symmetry of A. Using the operator Ψ _A , the addition ⊕ _A and the scalar multiplication ⊙ _A on ℝ F ( A ) can be defined by B ⊕ A C = Ψ A ( Ψ A - 1 ( B ) + Ψ A - 1 ( C ) ) , λ ⊙ A B = Ψ A ( λ · Ψ A - 1 ( B ) ) , where B , C ∈ ℝ F ( A ) and λ ∈ ℝ . Under these two operations, the triple ( ℝ F ( A ) , ⊕ A , ⊙ A ) constitutes a vector space on ℝ . Obviously, it is a linear subspace of ℝ F . So the triple ( ℝ F ( A ) , ⊕ A , ⊙ A ) is called the space of A-linearly correlated fuzzy numbers. Similarly, the subtraction between B and C can be defined by B ⊖ A C = Ψ A ( Ψ A - 1 ( B ) - Ψ A - 1 ( C ) ) . It is easy to verify that B ⊖ A C = B ⊕ A ( ( - 1 ) ⊙ A C ) . Moreover, the norm ∥ · ∥ Ψ A : ℝ F ( A ) → [ 0 , + ∞ ) is defined by ∥ B ∥ Ψ A = ∥ Ψ A - 1 ( B ) ∥ ∞ , where B ∈ ℝ F ( A ) , ∥ · ∥ _∞ stands for the infinity norm of 2-dimensional Euclidean space ℝ 2 .

Notice that the operator Ψ _A is a linear isomorphism between ℝ 2 and ℝ F ( A ) . Therefore, the completeness of ℝ 2 implies that the space ℝ F ( A ) of A-linearly correlated fuzzy numbers is a complete normed linear space of dimension 2. In addition, the metric d Ψ A : ℝ F ( A ) × ℝ F ( A ) → [ 0 , + ∞ ) induced by the norm ∥ · ∥  _{Ψ A} is given by d Ψ A ( B , C ) = ∥ B ⊖ A C ∥ Ψ A = ∥ Ψ A ( Ψ A - 1 ( B ) ⊖ A Ψ A - 1 ( C ) ) ∥ Ψ A = ∥ Ψ A - 1 ( B ) - Ψ A - 1 ( C ) ∥ ∞ , where B , C ∈ ℝ F ( A ) . The metric d _{Ψ A} has the following properties:

Proposition 3.1. Let B , C , D , E ∈ ℝ F ( A ) and λ ∈ ℝ . Then

(i) d _{Ψ A}  (B ⊕  _A D, C ⊕  _A D) = d _{Ψ A}  (B, C);

(ii) d _{Ψ A}  (λ ⊙  _A B, λ ⊙  _A C) = ∣ λ ∣ d _{Ψ A}  (B, C);

(iii) d _{Ψ A}  (B ⊕  _A D, C ⊕  _A E) ≤ d _{Ψ A}  (B, C) + d _{Ψ A}  (D, E).

Proof. (i) By the definition of the metric d _{Ψ A} , we have d Ψ A ( B ⊕ A D , C ⊕ A D ) = ∥ ( B ⊕ A D ) ⊖ A ( C ⊕ A D ) ∥ Ψ A = ∥ B ⊖ A C ∥ Ψ A = d Ψ A ( B , C ) . (ii) d Ψ A ( λ ⊙ A B , λ ⊙ A C ) = ∥ λ ⊙ A B ⊖ A λ ⊙ A C ∥ Ψ A = ∥ Ψ A - 1 ( λ ⊙ A B ) - Ψ A - 1 ( λ ⊙ A C ) ∥ ∞ = ∥ Ψ A - 1 ( Ψ A ( λ ⊙ Ψ A - 1 ( B ) ) ) - Ψ A - 1 ( Ψ A ( λ ⊙ Ψ A - 1 ( C ) ) ) ∥ ∞ = ∥ λ ⊙ Ψ A - 1 ( B ) - λ ⊙ Ψ A - 1 ( C ) ∥ ∞ = ∣ λ ∣ ∥ Ψ A - 1 ( B ) - Ψ A - 1 ( C ) ∥ ∞ = ∣ λ ∣ d Ψ A ( B , C ) . (iii) Note that d _{Ψ A} is a metric on ℝ F ( A ) . By the triangle inequality of d _{Ψ A} , we get d Ψ A ( B ⊕ A D , C ⊕ A E ) ≤ d Ψ A ( B ⊕ A D , C ⊕ A D ) + d Ψ A ( C ⊕ A D , C ⊕ A E ) = d Ψ A ( B , C ) + d Ψ A ( D , E ) , where the last equality we have used the result (i) of the proposition.□

3.2 A is symmetric

Let A ∈ ℝ F \ ℝ be a symmetric fuzzy number with the symmetry point x^∗. For any ( q , r ) ∈ ℝ 2 , by Theorem 3.6 in [5], the operator Ψ _A  (q, r) is generally not injective and its inverse image Ψ A - 1 ( Ψ A ( q , r ) ) = { ( q , r ) , ( - q , 2 qx ∗ + r ) } . Usually, the preimage of Ψ _A contains two elements unless q = 0. Then, we can introduce a binary relation on ℝ 2 .

Definition 3.1. Let A ∈ ℝ F \ ℝ be a symmetric fuzzy number with the symmetry point x^∗ and let ≡ _A be a binary relation on ℝ 2 . We say that the element (q₁, r₁) is in binary relation ≡ _A to the element (q₂, r₂), which is abbreviated as (q₁, r₁) ≡  _A  (q₂, r₂), if q₁ = q₂, r₁ = r₂ or q₁ = - q₂, r₁ = 2q₂x^∗ + r₂.

Proposition 3.2. ([12]). Let A ∈ ℝ F \ ℝ be a symmetric fuzzy number with the symmetry point x^∗. The binary relation ≡ _A is an equivalence relation on ℝ 2 .

Using the equivalence relation ≡ _A on ℝ 2 , for any ( q , r ) ∈ ℝ 2 , the equivalence class of (q, r) can be defined by [ ( q , r ) ] ≡ A = def { ( q , r ) , ( - q , 2 qx ∗ + r ) } .

Note that each equivalence relation contains at most two elements for any ( q , r ) ∈ ℝ 2 . If q = 0, then the equivalence class has only one element, i.e., [(0, r)] _{≡ A}  = {(0, r)}. If q ≠ 0, then the equivalence class has exactly two elements. In this case, the abscissas of these two elements are always a pair of opposite numbers.

Moreover, the quotient set of ℝ 2 under the equivalence relation ≡ _A is defined by ℝ 2 / ≡ A = def { [ ( q , r ) ] ≡ A | ( q , r ) ∈ ℝ 2 } .

Now we define the operator Ψ ˆ A : ℝ 2 / ≡ A → ℝ F , which associates an equivalence class [(q, r)] _{≡ A} in ℝ 2 / ≡ A to the fuzzy number Ψ ˆ A ( [ ( q , r ) ] ≡ A ) = qA + r in ℝ F . It is easy to see that Ψ ˆ A is a bijection from ℝ 2 / ≡ A to ℝ F ( A ) .

Without loss of generality, we assume that the first coordinate q ≥ 0 of the representative element [(q, r)] _{≡ A} of each equivalence class in ℝ 2 / ≡ A . Then, the addition ⊕ ˆ A and the scalar multiplication ⊙ ˆ A in ℝ 2 / ≡ A can be defined by [ ( q , r ) ] ≡ A ⊕ ^ A [ ( p , s ) ] ≡ A = d e f { ( q , r ) + ( p , s ) , ( − q , 2 q x ∗ + r ) + ( − p , 2 p x ∗ + s ) } = { ( q + p , r + s ) , ( − ( q + p ) , 2 ( q + p ) x ∗ + r + s ) } = [ ( q + p , r + s ) ] ≡ A , λ ⊙ ˆ A [ ( q , r ) ] ≡ A = def { λ · ( q , s ) , λ · ( - q , 2 qx ∗ + r ) } = { ( λ q , λ r ) , ( - λ q , 2 λ qx ∗ + λ r ) } = { [ ( λ q , λ r ) ] ≡ A , λ ≥ 0 , [ ( - λ q , 2 λ qx ∗ + λ r ) ] ≡ A , λ < 0 for all [ ( q , r ) ] ≡ A , [ ( p , s ) ] ≡ A ∈ ℝ 2 / ≡ A and λ ∈ ℝ .

Remark 1. In the definition of addition, it should be emphasized that we require that the addition can be carried out only for those elements whose first coordinate elements in the equivalence class have the same sign. In the definition of scalar multiplication, when λ < 0, we take (- λq, 2λqx₀ + λr) as the representative element to ensure that the first coordinate is nonnegative. Actually, two equivalence classes [(λq, λr)] _{≡ A} and [(- λq, 2λqx₀ + λr)] _{≡ A} are coincident.

Using the operator Ψ ˆ A and the addition and scalar multiplication defined on ℝ 2 / ≡ A , the corresponding addition ⊕ _A and the corresponding scalar multiplication ⊙ _A in ℝ F ( A ) can be defined by B ⊕ A C = Ψ ˆ A ( Ψ ˆ A - 1 ( B ) ⊕ ˆ A Ψ ˆ A - 1 ( C ) ) , λ ⊙ A B = Ψ ˆ A ( λ ⊙ ˆ A Ψ ˆ A - 1 ( B ) ) for all B , C ∈ ℝ F ( A ) and λ ∈ ℝ .

Proposition 3.3. ([12]). Let A ∈ ℝ F \ ℝ be a symmetric fuzzy number with the symmetry point x^∗. If B , C ∈ ℝ F ( A ) and λ ∈ ℝ , then

(i) B ⊕ A C ∈ ℝ F ( A ) ;

(ii) λ ⊙ A B ∈ ℝ F ( A ) .

By the addition ⊕ _A and the scalar multiplication ⊙ _A in ℝ F ( A ) , the subtraction ⊖ A between B and C in ℝ F ( A ) can be presented by B ⊖ A C = B ⊕ A ( ( - 1 ) ⊙ A C ) .

Proposition 3.4. ([12]). Let A ∈ ℝ F \ ℝ be a symmetric fuzzy number with the symmetry point x^∗. If B , C ∈ ℝ F ( A ) , λ , μ ∈ ℝ , then

(i) B ⊕  _A B = 2 ⊙  _A B,

(ii) B ⊖ A B = B ⊕ ( - B ) ,

(iii) B ⊖ A C = ( - 1 ) ⊙ A ( C ⊖ A B ) ,

(iv) ( B ⊕ A C ) ⊖ A C = B ⊕ A ( C ⊖ A C ) ,

(v) λ ⊙  _A  (B ⊕  _A C) = λ ⊙  _A  (B) ⊕  _A λ ⊙  _A  (C),

(vi) (λ + μ) ⊙  _A B = λ ⊙  _A B ⊕  _A μ ⊙  _A B, where λμ ≥ 0.

Remark 2. From Proposition 3.4, we know that the distributive law (λ + μ) ⊙  _A B = λ ⊙  _A B ⊕  _A μ ⊙  _A B holds only when λμ ≥ 0. Furthermore, it is easy to see that the triple ( ℝ F ( A ) , ⊕ A , ⊙ A ) can not form a linear space. Proposition 3.3 shows that the difference between any two fuzzy numbers in ℝ F ( A ) always exists, but B ⊖ A B is not equal to 0 unless B degenerates into an ordinary real number.

Proposition 3.5. Let A 1 , A 2 ∈ ℝ F \ ℝ be two symmetric fuzzy numbers with the symmetry points x 1 ∗ and x 2 ∗ , respectively. Suppose that q _i , p _i  ≥ 0 and r i , s i ∈ ℝ ( i = 1 , 2 ) such that Ψ ˆ A 1 ( [ ( q i , r i ) ] ≡ A ) = Ψ ˆ A 2 ( [ ( p i , s i ) ] ≡ A ) ( i = 1 , 2 ) . Then

(i) Ψ ˆ A 1 ( [ ( q 1 , r 1 ) ] ≡ A ) ⊕ A 1 Ψ ˆ A 1 ( [ ( q 2 , r 2 ) ] ≡ A )

= Ψ ˆ A 2 ( [ ( p 1 , s 1 ) ] ≡ A ) ⊕ A 2 Ψ ˆ A 2 ( [ ( p 2 , s 2 ) ] ≡ A ) ;

(ii) λ ⊙ A 1 Ψ ˆ A 1 ( [ ( q 1 , r 1 ) ] ≡ A )

= λ ⊙ A 2 Ψ ˆ A 2 ( [ ( p 1 , s 1 ) ] ≡ A ) , λ ∈ ℝ .

Proof. (i) Since q _i , p _i  ≥ 0  (i = 1, 2), it follows that q _i  = 0 is equivalent to p _i  = 0, i = 1, 2. If q₁ = p₁ = 0, then we can obtain r 1 = Ψ ˆ A 1 ( [ ( q 1 , r 1 ) ] ≡ A ) = Ψ ˆ A 2 ( [ ( p 1 , s 1 ) ] ≡ A ) = s 1 . Then, we have Ψ ˆ A 1 ( [ ( q 1 , r 1 ) ] ≡ A ) ⊕ A 1 Ψ ˆ A 1 ( [ ( q 2 , r 2 ) ] ≡ A ) = ( 0 A 1 + r 1 ) ⊕ A 1 Ψ ˆ A 1 ( [ ( q 2 , r 2 ) ] ≡ A = ( 0 A 2 + s 1 ) ⊕ A 2 Ψ ˆ A 2 ( [ ( p 2 , s 2 ) ] ≡ A = Ψ ˆ A 2 ( [ ( p 1 , s 1 ) ] ≡ A ) ⊕ A 2 Ψ ˆ A 2 ( [ ( p 2 , s 2 ) ] ≡ A ) , which implies that the result (i) holds true. If q _i  > 0 and p _i  > 0, i = 1, 2, then we can infer that q 1 A 1 + r 1 = p 1 A 2 + s 1 , q 2 A 1 + r 2 = p 2 A 2 + s 2 . Using the similar argument as Lemma 5 in [9], we conclude that the result (i) still holds.

(ii) If λ ≥ 0, it can be inferred that λ q 1 A 1 + λ r 1 = λ p 1 A 2 + λ s 1 , which leads to Ψ ˆ A 1 ( [ ( λ q 1 , λ r 1 ) ] ≡ A ) = Ψ ˆ A 2 ( [ ( λ p 1 , λ s 1 ) ] ≡ A ) . Therefore, we can obtain λ ⊙ A 1 Ψ ˆ A 1 ( [ ( q 1 , r 1 ) ] ≡ A ) = Ψ ˆ A 1 ( [ ( λ q 1 , λ r 1 ) ] ≡ A ) = Ψ ˆ A 2 ( [ ( λ p 1 , λ s 1 ) ] ≡ A ) = λ ⊙ A 2 Ψ ˆ A 2 ( [ ( p 1 , s 1 ) ] ≡ A ) . If λ < 0, then we have - λ q 1 A 1 + 2 λ q 1 x 1 ∗ + λ r 1 = λ p 1 A 2 + 2 λ p 1 x 2 ∗ + λ s 1 . Thus, we can obtain Ψ ˆ A 1 ( [ ( - λ q 1 , 2 λ q 1 x 1 ∗ + λ r 1 ) ] ≡ A ) = Ψ ˆ A 2 ( [ ( - λ p 1 , 2 λ p 1 x 2 ∗ + λ s 1 ) ] ≡ A ) . Therefore, it follows that λ ⊙ A 1 Ψ ˆ A 1 ( [ ( q 1 , r 1 ) ] ≡ A ) = Ψ ˆ A 1 ( [ ( - λ q 1 , 2 λ q 1 x 1 ∗ + λ r 1 ) ] ≡ A ) = Ψ ˆ A 2 ( [ ( - λ p 1 , 2 λ p 1 x 2 ∗ + λ s 1 ) ] ≡ A ) = λ ⊙ A 2 Ψ ˆ A 2 ( [ ( p 1 , s 1 ) ] ≡ A ) . □

Remark 3. From Proposition 3.5, it can be seen that the addition ⊕ _A and the scalar multiplication ⊙ _A of linearly correlated fuzzy numbers are independent of the choice of the basic symmetric fuzzy number A. This result can be regarded as a complement of Lemma 5 in [9].

4.1 A is non-symmetric

Since A is non-symmetric, for any B ∈ ℝ F ( A ) , there exists a unique pair of numbers ( q , r ) ∈ ℝ 2 such that B = Ψ _A  (q, r). Note that Ψ _A is a linear isomorphism between ℝ 2 and ℝ F ( A ) and ℝ 2 is a linear space. Then, we can introduce the notion of the LC-difference of A-linearly correlated fuzzy numbers.

Definition 4.1. Let A be a non-symmetric fuzzy number. For any B , C ∈ ℝ F ( A ) , the LC-difference ⊟ A of B and C is defined by B ⊟ A C = Ψ A ( q - p , r - s ) = ( q - p ) A + ( r - s ) , where (q, r) and (p, s) are two pairs of numbers such that B = Ψ _A  (q, r) and C = Ψ _A  (p, s), respectively.

Remark 4. The properties of the linear isomorphism Ψ _A implies Definition 4.1 and the difference introduced in subsection 3.1 are coincident. That is, B ⊟ A C = B ⊖ A C for any B , C ∈ ℝ F ( A ) .

4.2 A is symmetric

From subsection 3.2, when A is a symmetric fuzzy number, we know that Ψ ˆ A is a bijection from the quotient set ℝ 2 / ≡ A to ℝ F ( A ) . Taking into account the characteristic of the elements contained in the equivalence class in ℝ 2 / ≡ A , we first introduce the notion of the LC-difference on ℝ 2 / ≡ A .

Definition 4.2. Let A be a symmetric fuzzy number with the symmetry point x^∗ and let [ ( q , r ) ] ≡ A , [ ( p , s ) ] ≡ A ∈ ℝ 2 / ≡ A . The LC-difference ⊟ ˆ A of [(q, r)] _{≡ A} and [(p, s)] _{≡ A} is defined by [ ( q , r ) ] ≡ A ⊟ ˆ A [ ( p , s ) ] ≡ A = { [ ( q - p , r - s ) ] ≡ A , q ≥ p , [ ( p - q , 2 ( q - p ) x ∗ + r - s ) ] ≡ A , q < p .

From Definition 4.2, we will introduce the notion of the LC-difference in ℝ F ( A ) by using the bijection Ψ ˆ A .

Definition 4.3. Let A be a symmetric fuzzy number with the symmetry point x^∗. For any B , C ∈ ℝ F ( A ) , the LC-difference of B and C is defined by B ⊟ A C = Ψ ˆ A ( [ ( q , r ) ] ≡ A ⊟ ˆ A [ ( p , s ) ] ≡ A ) , where [ ( q , r ) ] ≡ A = Ψ ˆ A - 1 ( B ) and [ ( p , s ) ] ≡ A = Ψ ˆ A - 1 ( C ) .

Example 1. Let A = (0 ; 1 ;2) be a symmetric triangle fuzzy number with the symmetry point x^∗ = 1 and let B = (-1 ; 0 ;1), C = (3 ; 5 ;7). Obviously, we get B = A - 1 and C = 2A + 3. Furthermore, we can obtain B = Ψ ˆ A ( [ ( 1 , - 1 ) ] ≡ A ) , C = Ψ ˆ A ( [ ( 2 , 3 ) ] ≡ A ) . Then, we have B ⊟ A C = Ψ ˆ A ( [ ( 1 , - 1 ) ] ≡ A ⊟ ˆ A [ ( 2 , 3 ) ] ≡ A ) = Ψ ˆ A ( [ ( 1 , - 6 ) ] ≡ A = A - 6 = ( - 6 ; - 5 ; - 4 ) . C ⊟ A B = Ψ ˆ A ( [ ( 2 , 3 ) ] ≡ A ⊟ ˆ A [ ( 1 , - 1 ) ] ≡ A ) = Ψ ˆ A ( [ ( 1 , 4 ) ] ≡ A = A + 4 = ( 4 ; 5 ; 6 ) .

Proposition 4.1. Let A ∈ ℝ F \ ℝ be a symmetric fuzzy number with the symmetry point x^∗. If B , C ∈ ℝ F ( A ) , λ ∈ ℝ then (i) B ⊟ A B = 0 ;

(ii) B ⊟ A C = ( - 1 ) ⊙ A ( C ⊟ A B ) ;

(iii) ( B ⊕ A C ) ⊟ A C = B ;

(iv) B ⊟ A ( B ⊕ A C ) = ( - 1 ) ⊙ A C .

(v) λ ⊙ A ( B ⊟ A C ) = ( λ ⊙ A B ) ⊟ A ( λ ⊙ A C ) .

Proof. (i) By Definition 4.3, it is obvious.

(ii) Since B , C ∈ ℝ F ( A ) , there exist two pairs of numbers (q, r) and (p, s) with q, p ≥ 0 such that B = Ψ ˆ A ( [ ( q , r ) ] ≡ A ) , C = Ψ ˆ A ( [ ( p , s ) ] ≡ A ) . Without loss of generality, assume q ≥ p, by Definitions 4.2 and 4.3, it follows that B ⊟ A C = Ψ ˆ A ( [ ( q - p , r - s ) ] ≡ A ) , C ⊟ A B = Ψ ˆ A ( [ ( q - p , 2 ( p - q ) x ∗ + s - r ) ] ≡ A ) . Then, we can infer that ( - 1 ) ⊙ A ( C ⊟ A B ) = ( - 1 ) ⊙ A Ψ ˆ A ( [ ( q - p , 2 ( p - q ) x ∗ + s - r ) ] ≡ A ) = Ψ ˆ A ( ( - 1 ) ⊙ ˆ A [ ( q - p , 2 ( p - q ) x ∗ + s - r ) ] ≡ A ) = Ψ ˆ A ( [ ( q - p , r - s ) ] ≡ A ) = B ⊟ A C .

(iii) It is a direct consequence of Definitions 4.2 and 4.3.

(iv) From (ii) and (iii), we can obtain B ⊟ A ( B ⊕ A C ) = ( - 1 ) ⊙ A ( ( B ⊕ A C ) ⊟ A B ) = ( - 1 ) ⊙ A C .

(v) Similar to (ii), we assume B = Ψ ˆ A ( [ ( q , r ) ] ≡ A ) and C = Ψ ˆ A ( [ ( p , s ) ] ≡ A ) . The proof is divided into four cases.

Case 1: If λ ≥ 0 and q ≥ p, then we get λ ⊙ A ( B ⊟ A C ) = λ ⊙ A Ψ ˆ A ( [ ( q - p , r - s ) ] ≡ A ) = Ψ ˆ A ( [ ( λ q - λ p , λ r - λ s ) ] ≡ A ) = ( Ψ ˆ A ( [ ( λ q , λ r ) ] ≡ A ) ) ⊟ A ( Ψ ˆ A ( [ ( λ p , λ s ) ] ≡ A ) ) = ( λ ⊙ A Ψ ˆ A ( [ ( q , r ) ] ≡ A ) ) ⊟ A ( λ ⊙ A Ψ ˆ A ( [ ( p , s ) ] ≡ A ) ) = ( λ ⊙ A B ) ⊟ A ( λ ⊙ A C ) . Case 2: If λ ≥ 0 and q < p, then we obtain λ ⊙ A ( B ⊟ A C ) = λ ⊙ A Ψ ˆ A ( [ ( p - q , 2 ( q - p ) x ∗ + r - s ) ] ≡ A ) = Ψ ˆ A ( [ ( λ p - λ q , 2 λ ( q - p ) x ∗ + λ r - λ s ) ] ≡ A ) = Ψ ˆ A ( [ ( λ q , λ r ) ] ≡ A ) ⊟ A Ψ ˆ A ( [ ( λ p , λ s ) ] ≡ A ) = ( λ ⊙ A B ) ⊟ A ( λ ⊙ A C ) .

Case 3: If λ < 0 and q ≥ p, then we have λ ⊙ A ( B ⊟ A C ) = λ ⊙ A Ψ ˆ A ( [ ( q - p , r - s ) ] ≡ A ) = Ψ ˆ A ( [ ( λ ( p - q ) , 2 λ ( q - p ) x ∗ + λ r - λ s ) ] ≡ A ) = ( λ ⊙ A Ψ ˆ A ( [ ( q , r ) ] ≡ A ) ) ⊟ A ( λ ⊙ A Ψ ˆ A ( [ ( p , s ) ] ≡ A ) ) = ( λ ⊙ A B ) ⊟ A ( λ ⊙ A C ) .

Case 4: If λ < 0 and q < p, then we can infer that λ ⊙ A ( B ⊟ A C ) = λ ⊙ A Ψ ˆ A ( [ ( p - q , 2 ( q - p ) x ∗ + r - s ) ] ≡ A ) = Ψ ˆ A ( [ ( λ ( q - p ) , λ r - λ s ) ] ≡ A ) = ( λ ⊙ A Ψ ˆ A ( [ ( q , r ) ] ≡ A ) ) ⊟ A ( λ ⊙ A Ψ ˆ A ( [ ( p , s ) ] ≡ A ) ) = ( λ ⊙ A B ) ⊟ A ( λ ⊙ A C ) . □

Remark 5. For any B , C ∈ ℝ F ( A ) , the LC-difference ⊟ A and the difference ⊖ A of B and C exist, but, in generally, B ⊟ A C ≠ B ⊖ A C if A is a symmetric fuzzy number. It can be seen that the LC-difference has better properties from Propositions 3.4 and 4.1.

Now we introduce a corresponding norm ∥ · ∥ ∞ ˆ on the quotient set ℝ 2 / ≡ A by using the infinity norm ∥ · ∥ _∞ on ℝ 2 . For any [ ( q , r ) ] ≡ A ∈ ℝ 2 / ≡ A , we define ∥ [ ( q , r ) ] ≡ A ∥ ∞ ˆ = max { ∥ ( q , r ) ∥ ∞ , ∥ ( - q , 2 qx ∗ + r ) ∥ ∞ } = max { | q | , | r | , | 2 qx ∗ + r | } . Accordingly, the norm ∥ · ∥ Ψ ˆ A on ℝ F ( A ) can be defined by ∥ · ∥ Ψ ˆ A = ∥ Ψ ˆ A - 1 ( B ) ∥ ∞ ˆ , ∀ B ∈ ℝ F ( A ) . It should be noted that ( ℝ F ( A ) , ∥ · ∥ Ψ ˆ A ) is only a normed space, not a normed linear space. Using the norm ‖ · ‖ Ψ ^ A on ℝ F ( A ) , the induced metric d Ψ ˆ A on ℝ F ( A ) can be defined by d Ψ ˆ A ( B , C ) = ∥ B ⊟ A C ∥ Ψ ˆ A .

Remark 6. According to the definition of the metric d Ψ ˆ A , it is easy to verify that the operator Ψ ˆ A is an isometric isomorphism from ℝ 2 / ≡ A to ℝ F ( A ) . Since the quotient set ℝ 2 / ≡ A as a metric subspace is complete, we can infer that the metric space ℝ F ( A ) is also complete.

Proposition 4.2. Let A ∈ ℝ F \ ℝ be a symmetric fuzzy number with the symmetry point x^∗. If B , C , D , E ∈ ℝ F ( A ) , λ , μ ∈ ℝ , then

(i) d Ψ ˆ A ( B , C ) = d Ψ ˆ A ( B ⊟ A C , 0 ) ;

(ii) d Ψ ˆ A ( λ ⊙ A B , λ ⊙ A C ) = | λ | d Ψ ˆ A ( B , C ) ;

(iii) d Ψ ˆ A ( B ⊕ A D , C ⊕ A D ) = d Ψ ˆ A ( B , C ) ;

(iv) d Ψ ˆ A ( B ⊕ A D , C ⊕ A E ) ≤ d Ψ ˆ A ( B , C ) + d Ψ ˆ A ( D , E ) .

Proof. (i) Using the definition of the metric d Ψ ˆ A , it is obvious.

(ii) By Proposition 4.1, we can obtain

d Ψ ˆ A ( λ ⊙ A B , λ ⊙ A C ) = ∥ ( λ ⊙ A B ) ⊟ A ( λ ⊙ A C ) ∥ Ψ ˆ A = ∥ λ ⊙ A ( B ⊟ A C ) ∥ Ψ ˆ A = | λ | ∥ B ⊟ A C ∥ Ψ ˆ A = | λ | d Ψ ˆ A ( B , C ) . (iii) Assume B = Ψ ˆ A ( [ ( q , r ) ] ≡ A ) , C = Ψ ˆ A ( [ ( p , s ) ] ≡ A ) , D = Ψ ˆ A ( [ ( w , v ) ] ≡ A ) . If q ≥ p, then we have d Ψ ˆ A ( B ⊕ A D , C ⊕ A D ) = ∥ ( B ⊕ A D ) ⊟ A ( C ⊕ A D ) ∥ Ψ ˆ A = ∥ B ⊟ A C ∥ Ψ ˆ A = d Ψ ˆ A ( B , C ) . If q < p, then it follows from Proposition 4.1 that ( B ⊕ A D ) ⊟ A ( C ⊕ A D ) = ( - 1 ) ⊙ A ( ( C ⊕ A D ) ⊟ A ( B ⊕ A D ) ) = ( - 1 ) ⊙ A ( C ⊟ A B ) . By (i) and (ii), we can obtain d Ψ ˆ A ( B ⊕ A D , C ⊕ A D ) = d ( ( - 1 ) ⊙ A ( C ⊟ A B ) , 0 ) = d Ψ ˆ A ( B , C ) .

(iv) Using the triangle inequality and (iii), the proof is similar to (iv) of Proposition 3.1.□

4.3 LC-difference of interval numbers

As a special case of symmetric fuzzy numbers, in this part, we will examine the LC-difference of interval numbers.

For any closed interval [a^-, a⁺], equivalently, we can treat it as a symmetric fuzzy number with the membership function A (x), where A (x) =1 if x ∈ [a^-, a⁺], A (x) =0 if x ∉ [a^-, a⁺]. For simplicity, we denote by I ℝ the set of all closed intervals on ℝ represented by the corresponding membership function. The length of A is denoted by len (A) = a⁺ - a^-. In fact, for any α ∈ [0, 1], the α-level set [A]  _α = [a^-, a⁺]. For intuitiveness, we still use the symbol [a^-, a⁺] to represent the interval number A. Obviously, I ℝ ⊂ ℝ F . Thus, for any A ∈ I ℝ , it is a symmetric fuzzy number with the symmetry point x A ∗ = a - + a + 2 .

Proposition 4.3. Let A ∈ I ℝ \ ℝ be an interval number with the symmetry point x A ∗ . For any B ∈ I ℝ , there exists a pair of numbers ( q , r ) ∈ ℝ 2 with q ≥ 0 such that B = qA + r.

Proof. Assume len (A) = a⁺ - a^-, len (B) = b⁺ - b^-. Since A ∈ I ℝ \ ℝ , it follows that len (A) >0. Take q = len ( B ) len ( A ) and r = x B ∗ - qx A ∗ , where x B ∗ = b - + b + 2 . Clearly, q ≥ 0. Then, for any α ∈ [0, 1], we can obtain q [ A ] α + r = q [ a - , a + ] + r = [ qa - + r , qa + + r ] = [ b - , b + ] = [ B ] α , which implies B = qA + r. □

Corollary 4.1. Let A ∈ I ℝ \ ℝ be an interval number with the symmetry point x A ∗ . For a given B ∈ I ℝ , if (q, r) is a pair of numbers with q ≥ 0 such that B = qA + r, then B = - qA + 2 qx A ∗ + r .

Proof. According to the statement in section 3.2, it is a direct consequence of Proposition 4.3. □

Example 2. Let A = [-1, 1] and B = [2, 6] be two interval numbers. Then we have x A ∗ = 0 , x B ∗ = 4 , len (A) =2 and len (B) =4. By Proposition 4.3, we take q = 2, r = x B ∗ - qx A ∗ = 4 . So we get B = 2A + 4. Meantime, we also obtain B = -2A + 4 by Corollary 4.4.

Based on Proposition 4.3 and Corollary 4.4, we can explicitly obtain the notion of the LC-difference between two interval numbers.

Definition 4.4. Let A ∈ I ℝ \ ℝ be an interval number with the symmetry point x A ∗ and let B , C ∈ I ℝ be two interval numbers. Suppose that (q, r) and (p, s) are two pairs of numbers with q, p ≥ 0 such that B = qA + r and C = pA + s. Then the LC-difference of B and C is given by B ⊟ A C = Ψ ˆ A ( [ ( q , r ) ] ≡ A ⊟ ˆ A [ ( p , s ) ] ≡ A ) = { ( q - p ) A + ( r - s ) , q ≥ p , ( p - q ) A + 2 ( q - p ) x A ∗ + r - s , q < p .

Example 3. Let A = [0, 2] be an interval number with the symmetry point x A ∗ = 1 and let B = [-2, 4], C = [1, 3]. According to Proposition 4.3 and Corollary 4.4, we can obtain B = 3 A - 2 = - 3 A + 4 , C = A + 1 = - A + 3 . Thus, we have [ ( 3 , - 2 ) ] ≡ A = { ( 3 , - 2 ) , ( - 3 , 4 ) } , [ ( 1 , 1 ) ] ≡ A = { ( 1 , 1 ) , ( - 1 , 3 ) } . By Definition 4.4, it follows that B ⊟ A C = Ψ ˆ A ( [ ( 2 , - 3 ) ] ≡ A ) = 2 A - 3 = [ - 3 , 1 ] , C ⊟ A B = Ψ ˆ A ( [ ( 2 , - 1 ) ] ≡ A ) = 2 A - 1 = [ - 1 , 3 ] .