A fuzzy soft set based approximate reasoning method 1

Abstract

Soft set theory, proposed by Molodtsov, has been regarded as an effective mathematical tool for dealing with uncertainties. This paper is devoted to the discussion of fuzzy soft set based approximate reasoning. First, based on fuzzy implication operators, the notion of fuzzy soft implication relation between fuzzy soft sets is introduced. The composition method of fuzzy soft implication relations is provided. Second, Triple I methods for fuzzy soft modus ponens (FSMP)and fuzzy soft modus tollens (FSMT) are investigated. Computational formulas for FSMP and FSMT with respect to left-continuous t-norms and its residual implication are presented. At last, the reversibility properties of Triple I methods are analyzed.

Keywords

Fuzzy set soft set fuzzy soft implication relation triple I method left-continuous t-norm

1 Introduction

To solve complicated problems in economics, engineering, environmental science and social science, methods in classical mathematics are not always successful because of various types of uncertainties presented in these problems. While probability theory, fuzzy set theory [38], rough set theory [23, 24], and other mathematical tools are well-known and often useful approaches to describing uncertainty, each of these theories has its inherent difficulties as pointed out in [20, 21]. In 1999, Molodtsov [20] introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainties. It has been proven useful in many fields such as decision making, data analysis, forecasting and texture classification.

Works on soft set theory are progressing rapidly. Maji et al. [12] and Ali et al. [1] defined several operations on soft sets and made a theoretical study on the theory of soft sets. Qin et al. [25] introduced the notion of soft equality and established lattice structures and soft quotient algebras of soft sets. Xiao et al. [33] proposed the notion of the exclusive disjunctive soft set and applied it to attribute reduction of incomplete information system. Babitha et al. [2] proposed the notion of soft set relation and many related concepts such as equivalent soft set relation, composition of soft set relations and soft set functions are discussed. It extends the notions of relations and functions to the framework of soft sets. Also the same authors [3] introduced the notions of antisymmetric relation and transitive closure of a soft set relation. An algorithm is presented for calculating the transitive closure of a soft set relation. Qin [27] introduced the notion of soft relation which is a generalization of the notion of soft set relation [2]. Furthermore, the connections between soft relations and fuzzy sets are analyzed. It is shown that fuzzy relation and fuzzy soft set may be considered as special cases of soft relation. Yang and Guo [37] proposed the notions of anti-reflexive kernel, symmetric kernel, reflexive closure, and symmetric closure of a soft set relation. Maji et al. [13] initiated the study on hybrid structures involving soft sets and fuzzy sets. They proposed the notion of fuzzy soft set as a fuzzy generalization of classical soft sets and some basic properties were discussed. Afterwards, many researchers have worked on this concept. Various kinds of extended fuzzy soft sets such as generalized fuzzy soft sets [17], intuitionistic fuzzy soft sets [14, 15], interval-valued fuzzy soft sets [36], vague soft sets [34], interval-valued intuitionistic fuzzy soft sets [9] and soft interval set [26, 41] were presented. The combination of soft set and rough set is another interesting topic [6 , 40]. Majumdar et al. [16, 18] initiated the study of uncertainty measures of soft sets and fuzzy soft sets. Some similarity measures between soft sets and fuzzy soft sets were presented. Kharal [11] introduced some set operation based distance and similarity measures for soft sets. Jiang [10] proposed some distance measures between intuitionistic fuzzy soft sets, and constructed some entropy measures on intuitionistic fuzzy soft sets and interval-valued fuzzy soft sets.

Approximate reasoning is one of the most important topic for a theory dealing with uncertainty. After Zadeh introduced the notion of fuzzy sets, various methods of fuzzy reasoning have been presented and they have been used as formal mathematical tools for reasoning under vagueness. We note that, up to now, approximate reasoning in the framework of soft set theory have not been conducted. The aim of the paper is to present a novel approach to fuzzy soft set based inference. The paper is organized as follows: In Section 2, we recall some notions and properties of soft sets and fuzzy soft sets. In Section 3, we propose the notion of fuzzy soft implication relation based on fuzzy implication operators. The composition of fuzzy soft implication relations and the composition of fuzzy soft implication relation and fuzzy soft set are introduced. In Section 4, Triple I methods for fuzzy soft modus ponens (FSMP)and fuzzy soft modus tollens (FSMT) are investigated. Computational formulas for both FSMP and FSMT with respect to left-continuous t-norms and its residual implication are obtained. Furthermore, the reversibility properties of Triple I method are analyzed. In Section 5, an application of triple I method for FSMP in medical diagnosis is presented. The paper is completed with some concluding remarks.

2 Soft sets and fuzzy soft sets

In this section, we recall some fundamental notions of soft sets and fuzzy soft sets [13 , 38].

Let U be an initial universe set and E the set of all possible parameters under consideration with respect to U. Usually, parameters are attributes, characteristics, or properties of objects in U. Molodtsov [20] defined the notion of soft set in the following way:

Definition 1. [20] A pair (F, A) is called a soft set over U, where A ⊆ E and F is a mapping given by F : A → P (U).

In other words, a soft set over U is a parameterized family of subsets of U. For e ∈ A, F (e) may be considered as the set of e-approximate elements of the soft set (F, A). For illustration, we consider the following example.

Example 1. Suppose that there are six houses in the universe U given by U = {h₁, h₂, h₃, h₄, h₅, h₆} and E = {e₁, e₂, e₃, e₄, e₅} is the set of parameters. Where e₁, e₂, e₃, e₄ and e₅ stand for the parameters ‘expensive’, ‘beautiful’, ‘wooden’, ‘cheap’ and ‘in the green surroundings’ respectively.

In this case, to define a soft set means to point out expensive houses, beautiful houses, and so on. The soft set (F, E) may describe the ‘attractiveness of the houses’ which Mr.X is going to buy. Suppose that F (e₁) = {h₂, h₄}, F (e₂) = {h₁, h₃}, F (e₃) = {h₃, h₄, h₅}, F (e₄) = {h₁, h₃, h₅}, F (e₅) = {h₁}. Then the soft set (F, E) is a parameterized family {F (e_i) ;1 ≤ i ≤ 5} of subsets of U and give us a collection of approximate descriptions of an object.

The theory of fuzzy sets initiated by Zadeh [38] provides an appropriate framework for representing and processing vague concepts by allowing partial memberships. Let U be a nonempty set, called universe. A fuzzy subset μ of U is defined by a membership function μ : U → [0, 1]. For x ∈ U, the membership value μ (x) essentially specifies the degree to which x belongs to the fuzzy subset μ. With the min-max system proposed by Zadeh [38], fuzzy set intersection, union and complement are defined componentwise as follows: $\begin{matrix} (μ \cap ν) (x) = μ (x) \land ν (x), \\ (μ \cup ν) (x) = μ (x) \lor ν (x), \\ μ^{c} (x) = 1 - μ (x), \end{matrix}$ where μ, ν are fuzzy subsets of U and x ∈ U. μ is called a fuzzy subset of ν if μ (x) ≤ ν (x) for any x ∈ U. In what follows, we denote by F (U) the set of all fuzzy subsets of U.

In most cases, the parameters of a soft space are fuzzy words or sentences involving fuzzy words. Considering this point, Maji et al. [13] introduced the concept of fuzzy soft set by combining soft set and fuzzy set.

Definition 2. [13] Let (U, E) be a soft space. A pair (F, A) is called a fuzzy soft set over U, where A ⊆ E and F is a mapping given by F : A → F (U).

Definition 3. [13] For two fuzzy soft sets (F, A) and (G, B) over a common universe U, we say that (F, A) is a fuzzy soft subset of (G, B), denoted by (F, A) ⊆ (G, B), if A ⊆ B and F (e) ⊆ G (e) for each e ∈ A.

Clearly, the notion of fuzzy soft set is a generalization of soft set.

3 Fuzzy soft implication relations

Our knowledge of complex systems is often incomplete, and therefore, we have to rely on the expert’s statements. Since a large part of expert knowledge consists of if-then statements, it is therefore important to formalize implication in fuzzy logic. The formalization is called fuzzy implication operator which extends the classic binary implication operator. In addition to its use in multivalued logic it plays an important role in fuzzy logic and Zadeh’s theory of approximate reasoning and computing with words [35].

A fuzzy implication operator is a binary operation → : [0, 1] ² → [0, 1] satisfying conditions [35]:

0 → a = 1.

a → 1 =1.

1 → 0 =0.

a → b is increasing with respect to b and decreasing with respect to a.

In addition to these a number of other desirable properties associated with fuzzy implication operator have also been suggested [4]. Triangular norms (t-norms) are closely related to fuzzy implication operators. In what follows, the least upper bound (greatest lower bound) of a subset G of [0, 1] will be denoted by ∨G (∧G), alternatively. A function ⊗ : [0, 1] ² → [0, 1] is said to be a t-norm if ⊗ is associative, commutative and satisfy the conditions a ⊗ 1 = a and that a ≤ b implies a ⊗ c ≤ b ⊗ c for all a, b, c ∈ [0, 1]. A t-norm ⊗ is left-continuous if a ⊗ ∨ {b_i ; i ∈ I} = ∨ {a ⊗ b_i ; i ∈ I} holds where a, b_i ∈ [0, 1] (i ∈ I), and I is a nonempty index set.

Theorem 1. [32] Suppose that ⊗ is a left-continuous t-norm, define →_⊗ : [0, 1] ² → [0, 1] as a → _⊗b = ∨ {x ∈ [0, 1] ; a ⊗ x ≤ b} for all a, b ∈ [0, 1]. Then →_⊗ is a fuzzy implication operator and

a → _⊗b = 1 iff a ≤ b.

a ≤ b → _⊗c iff b ≤ a → _⊗c.

a → _⊗ (b → _⊗c) = b → _⊗ (a → _⊗c).

1 → _⊗a = a.

a → _⊗ ∧ {b_i ; i ∈ I} = ∧ {a → _⊗b_i ; i ∈ I}.

∨ {b_i ; i ∈ I} → _⊗a = ∧ {b_i → _⊗a ; i ∈ I}.

a ⊗ b ≤ c iff a ≤ b → _⊗c.

Example 2. [32] The following are four left-continuous t-norms: $a \otimes_{L} b = (a + b - 1) \lor 0$ (1) $a \otimes_{G} b = a \land b$ (2) $a \otimes_{π} b = ab$ (3) $a \otimes_{0} b = {\begin{matrix} a \land b, & if a + b > 1, \\ 0, & if a + b \leq 1 . \end{matrix}$ (4) The implication operators corresponding to thet-norms ⊗_L, ⊗_G, ⊗_π and ⊗₀ are as follows (they are called Lukasiewicz operator, Godel operator, product operator, and R₀ operator, respectively): $a \to_{\otimes_{L}} b = (1 - a + b) \land 1$ (5) $a \to_{\otimes_{G}} b = {\begin{matrix} 1, & if a \leq b, \\ b, & if a > b . \end{matrix}$ (6) $a \to_{\otimes_{π}} b = {\begin{matrix} 1, & if a = 0, \\ \frac{b}{a} \land 1, & if a > 0 . \end{matrix}$ (7) $a \to_{\otimes_{0}} b = {\begin{matrix} 1, & if a \leq b, \\ (1 - a) \lor b, & if a > b . \end{matrix}$ (8)

Babitha and Sunil [2] introduced the notions of soft set relation and soft set function between soft sets. It extends the notions of classic binary relations and functions to the framework of soft sets.

Definition 4. [2] Let (F, A) and (G, B) be two soft sets over U.

The Cartesian product of (F, A) and (G, B) is defined as (F, A) × (G, B) = (H, A × B), where H : A × B → P (U × U) is given by H (a, b) = F (a) × G (b) for all (a, b) ∈ A × B.

A soft set relation from (F, A) to (G, B) is a soft subset of (F, A) × (G, B).

By this definition, a soft set relation from (F, A) to (G, B) is a soft set (H, C), where C ⊆ A × B and H (a, b) ⊆ F (a) × G (b) for every (a, b) ∈ C. Now, we extend this notion to fuzzy soft implication relation.

Definition 5. Let (F, A) and (G, B) be fuzzy soft sets over the universe U and V respectively. A fuzzy soft implication relation from (F, A) to (G, B) is a fuzzy soft set (H, C) over U × V, where C ⊆ A × B, H : C → F (U × V) is given by $H (a, b) (x, y) = F (a) (x) \to G (b) (y)$ for all (a, b) ∈ C, (x, y) ∈ U × V, and → is a fuzzy implication operator.

From this definition, (H, C) is a fuzzy soft implication relation from (F, A) to (G, B) if H (a, b) = F (a) → G (b) is a fuzzy implication from fuzzy subsets F (a) to G (b) for every (a, b) ∈ C. Clearly, if A = {a}, B = {b} are parameter sets with single element, then the fuzzy soft implication relation degenerate to fuzzy implication relation. For illustration, we consider the following example.

Example 3. We consider multiple rule single premise fuzzy inference model. Suppose that A₁ → B₁, A₂→ B₂, ⋯ and A_m → B_m are the inference rules, where A₁, A₂, ⋯ , A_m and B₁, B₂, ⋯ , B_m are fuzzy subsets of the universe U and V respectively. The antecedents of the rules forms a fuzzy soft set (F, M) over U, where M = {1, 2, ⋯ , m} and F (i) = A_i. Similarly, the consequents of the rules forms a fuzzy soft set (G, M) over V, where G (i) = B_i. The inference rules forms a fuzzy soft implication relation (H, C) from (F, M) to (G, M), where C = {(i, i) ;1 ≤ i ≤ m} ⊆ M × M, and H (i, i) = A_i → B_i is a fuzzy implication relation from U to V, i.e., → is a fuzzy implication operator, H (i, i) (x, y) = A_i (x) → B_i (y) for each (x, y) ∈ U × V.

This example shows that the inference rules of multiple rule single premise fuzzy inference model can be expressed as a fuzzy soft implication relation.

In some cases in uncertainty reasoning, we need to discuss the composition of inference rules. Accordingly, based on [27], we introduce the composition of fuzzy soft implication relations.

Definition 6. Let (F, A), (G, B) and (H, C) be fuzzy soft sets over U, V and W respectively, R = (R, D) a fuzzy soft implication relation from (F, A) to (G, B) and S = (S, K) a fuzzy soft implication relation from (G, B) to (H, C). The composition of R and S, denoted by S ○ R, is a fuzzy soft implication relation S ○ R = (L, K ○ D) from (F, A) to (H, C) given by: $\begin{matrix} K \circ D & = & {(a, c) \in A \times C; \\ \exists b \in B ((a, b) \in D \land (b, c) \in K)}, \\ L (a, c) (x, z) & = & \lor_{b \in E (a, c)} \lor_{y \in V} (S (b, c) (y, z) \\ \land R (a, b) (x, y)), \end{matrix}$ for each (a, c) ∈ K ∘ D and (x, z) ∈ U × W, where E (a, c) = {b ∈ B ; (a, b) ∈ D, (b, c) ∈ K} .

In this definition, K ∘ D is the composition of classical relations D and K. For each (a, c) ∈ K ∘ D and b ∈ E (a, c), by (a, b) ∈ D and (b, c) ∈ K we know that R (a, b) is a fuzzy implication relation from U to V and S (b, c) is a fuzzy implication relation from V to W. From fuzzy set theory, we know that ∨_y∈V (S (b, c) (y, z) ∧ R (a, b) (x, y)) = S (b, c) ∘ R (a, b) is the composition of fuzzy implication relations R (a, b) and S (b, c). Thus, L can be expressed as $L (a, c) = \cup_{b \in E (a, c)} S (b, c) \circ R (a, b),$ for each (a, c) ∈ K ∘ D.

Definition 7. Let (F, A) , (F^∗, A) and (G, B) , (G^∗, B) be fuzzy soft sets over U and V respectively, (H, C) a fuzzy soft implication relation from (F, A) to (G, B).

(1) The composition of (H, C) and (F^∗, A), denoted by (H, C) ∘ (F^∗, A), is a fuzzy soft set (G′, B′) over V given by: $B^{'} = {b \in B; \exists a \in A ((a, b) \in C)},$

G′ (b) (y) = ∨ _a∈E(b) ∨ _x∈U (H (a, b) (x, y) ∧ F^∗ (a) (x)) , for each b ∈ B′ and y ∈ V, where E (b) = {a ∈ A ; (a, b) ∈ C}.

(2) The composition of (H, C) and (G^∗, B), denoted by (H, C) ∘ (G^∗, B), is a fuzzy soft set (F′, A′) over U given by: $A^{'} = {a \in A; \exists b \in B ((a, b) \in C)},$

F′ (a) (x) = ∨ _b∈E(a) ∨ _y∈V (H (a, b) (x, y) ∧ G^∗ (b) (y)) , for each a ∈ A′ and x ∈ U, where E (a) = {b ∈ B ; (a, b) ∈ C}.

4 The triple I method for FSMP and FSMT

In this section, we present a fuzzy soft set based approximate reasoning method.

Since the fuzzy control achieved successful application in various fields, fuzzy reasoning as the basis of fuzzy control has gained much more attention by scholars. Fuzzy reasoning is based on fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT). FMP model can be expressed as:

Major premise:	A → B
Minor premise:	A ^∗
Conclusion:	B ^∗

Where A, A^∗ and B, B^∗ are fuzzy sets over U and V, respectively.

For FMP model, Zadeh [39] proposed the Compositional Rule of Inference (CRI). Despite CRI successes in various rule-based system application, its underlying semantic is unclear. Wang [31] proposed Triple I method and put the fuzzy inference in the framework of fuzzy logic. The main idea of Triple I method is that the minor premise A^∗ implies B^∗ should be fully supported by the major premise A → B. By using logic terminology, the expression (A (x) → B (y)) → (A^∗ (x) → B^∗ (y)) should be a tautology, i.e., for each x ∈ U, y ∈ V, the following condition holds: $(A (x) \to B (y)) \to (A^{*} (x) \to B^{*} (y)) = 1 .$

Now, we consider a fuzzy soft set based inference model, namely, fuzzy soft modus ponens (briefly, FSMP):

Major premise:	(F, A) → (G, B)
Minor premise:	(F^∗, A) (9)
Conclusion:	(G^∗, B)

Where (F, A), (F^∗, A) and (G, B), (G^∗, B) are fuzzy soft sets over U and V, respectively. Motivated by Wang’s triple I principle, we propose the following principle for FSMP.

Principle of triple I for FSMP: The FSMP conclusion (G^∗, B) of (9) is the smallest fuzzy soft set over V satisfying

$\begin{matrix} (F (α) (x) \to G (β) (y)) \\ \to (F^{*} (α) (x) \to G^{*} (β) (y)) = 1 \end{matrix}$ (10) for each α ∈ A, β ∈ B, x ∈ U, y ∈ V.

Theorem 2. Suppose that ⊗ is a left-continuous t-norm and the implication operator in FSMP is →_⊗. Then the Triple I solution of (9) can be expressed as follows: for each β ∈ B, y ∈ V,

$\begin{matrix} G^{*} (β) (y) = \sup_{α \in A} \sup_{x \in U} F^{*} (α) (x) \\ \otimes (F (α) (x) \to_{\otimes} G (β) (y)) . \end{matrix}$ (11)

Proof. First, we prove that (G^∗, B) determined by (11) satisfies (10). For any β ∈ B, y ∈ V, α ∈ A, x ∈ U, it follows from (11) that $\begin{matrix} G^{*} (β) (y) \geq F^{*} (α) (x) \otimes (F (α) (x) \to_{\otimes} G (β) (y)) . \end{matrix}$ By Theorem 1(7) we have F (α) (x) → _⊗G (β) (y) ≤ F^∗ (α) (x) → _⊗G^∗ (β) (y) and consequently $\begin{matrix} (F (α) (x) \to_{\otimes} G (β) (y)) \\ \to_{\otimes} (F^{*} (α) (x) \to_{\otimes} G^{*} (β) (y)) = 1 \end{matrix}$ by Theorem 1(1).

Next, we will verify that (G^∗, B) is the smallest fuzzy soft set over V satisfying (10). Assume that the fuzzy soft set (H, B) satisfies (10), that is $\begin{matrix} (F (α) (x) \to_{\otimes} G (β) (y)) \\ \to_{\otimes} (F^{*} (α) (x) \to_{\otimes} H (β) (y)) = 1 \end{matrix}$ for any β ∈ B, y ∈ V, α ∈ A, x ∈ U. It follows that $\begin{matrix} F (α) (x) \to_{\otimes} G (β) (y) \leq F^{*} (α) (x) \to_{\otimes} H (β) (y), \\ F^{*} (α) (x) \otimes (F (α) (x) \to_{\otimes} G (β) (y)) \leq H (β) (y), \end{matrix}$ and hence $\begin{matrix} G^{*} (β) (y) = \sup_{α \in A} \sup_{x \in U} F^{*} (α) (x) \otimes (F (α) (x) \\ \to_{\otimes} G (β) (y)) \leq H (β) (y) \end{matrix}$ Consequently, (G^∗, B) ⊆ (H, B) as required.

Corollary 1. Suppose that the implication operator in FSMP is →_{⊗_L}. Then the Triple I solution of (9) can be expressed as follows: for each β ∈ B, y ∈ V,

$\begin{matrix} G^{*} (β) (y) & = & \sup_{α \in A} \sup_{x \in E_{(α, β, y)}} F^{*} (α) (x) \\ \land (F^{*} (α) (x) + G (β) (y) - F (α) (x)) \end{matrix}$ (12)

where E_(α,β,y) = {x ∈ U ; F^∗ (α) (x) >0, F^∗ (α) (x) + G (β) (y) - F (α) (x) >0}.

Proof. If x ∈ E_(α,β,y), then F^∗ (α) (x) >0, F^∗ (α) (x) + G (β) (y) - F (α) (x) >0 and hence $\begin{matrix} F^{*} (α) (x) \otimes_{L} (F (α) (x) \to_{\otimes_{L}} G (β) (y)) \\ = (F^{*} (α) (x) + (1 - F (α) (x) + G (β) (y)) \land 1 - 1) \lor 0 \\ = ((F^{*} (α) (x) + G (β) (y) - F (α) (x)) \land F^{*} (α) (x)) \lor 0 \\ = (F^{*} (α) (x) + G (β) (y) - F (α) (x)) \land F^{*} (α) (x) . \end{matrix}$ Assume that x ∉ E_(α,β,y), then F^∗ (α) (x) =0 or F^∗ (α) (x) + G (β) (y) - F (α) (x) ≤0. If F^∗ (α) (x) =0, then F^∗ (α) (x) ⊗ _L (F (α) (x) → _{⊗_L}G (β) (y)) =0. If F^∗ (α) (x) + G (β) (y) - F (α) (x) ≤0, then $\begin{matrix} F^{*} (α) (x) \otimes_{L} (F (α) (x) \to_{\otimes_{L}} G (β) (y)) \\ = ((F^{*} (α) (x) + G (β) (y) - F (α) (x)) \land F^{*} (α) (x)) \lor 0 \\ = (F^{*} (α) (x) + G (β) (y) - F (α) (x)) \lor 0 = 0 . \end{matrix}$ It follows from (11) that $\begin{matrix} G^{*} (β) (y) = \sup_{α \in A} \sup_{x \in E_{(α, β, y)}} F^{*} (α) (x) \\ \land (F^{*} (α) (x) + G (β) (y) - F (α) (x)) . \end{matrix}$ Similarly, we have

Corollary 2. Suppose that the implication operator in FSMP is Godel implication operator →_{⊗_G}. Then the Triple I solution of (9) can be expressed as follows: for each β ∈ B, y ∈ V,

$\begin{matrix} G^{*} (β) (y) = \sup_{α \in A} \sup_{x \in U} F^{*} (α) (x) \\ \land (F (α) (x) \to_{\otimes_{G}} G (β) (y)) . \end{matrix}$ (13)

By this corollary, we know that with respect to Godel t-norm ⊗_G and Godel implication operator →_{⊗_G}, the Triple I solution for FSMP is just the composition of fuzzy soft implication relation F (α) → _{⊗_G}G (β) and fuzzy soft set (F^∗, A) in the sense of Definition 7. If A = {a}, B = {b} are parameter sets with single element, then the formula (13) degenerate to: $\begin{matrix} G^{*} (b) (y) = \sup_{x \in U} F^{*} (a) (x) \\ \land (F (a) (x) \to_{\otimes_{G}} G (b) (y)) . \end{matrix}$ This is Zadel’s CRI solution of FMP except that the implication operator is the Godel’s operator.

Corollary 3. Suppose that the implication operator in FSMP is →_⊗₀, then the Triple I solution of (9) can be expressed as follows: for each β ∈ B, y ∈ V,

$\begin{matrix} G^{*} (β) (y) & = & \sup_{α \in A} \sup_{x \in E_{(α, β, y)}} F^{*} (α) (x) \\ \land (F (α) (x) \to_{\otimes_{0}} G (β) (y)) \end{matrix}$ (14)

where E_(α,β,y) = {x ∈ U ; F^∗ (α) (x) + (F (α) (x) → _⊗₀G (β) (y)) >1}.

Theorem 3. Suppose that ⊗ is a left-continuoust-norm, the implication operator in FSMP is →_⊗, and there exists α ∈ A such that F (α) is a normal fuzzy subset of U (i.e., there exists x ∈ U such that F (α) (x) =1). Then the Triple I method for FSMP is reversible, i.e., if the fuzzy soft set (F^∗, A) in (9) equals (F, A), then (G^∗, B) equals (G, B).

Proof. Suppose that (F^∗, A) = (F, A) in (9), and α₀ ∈ A, x₀ ∈ U such that F (α₀) (x₀) =1. For each β ∈ B and y ∈ V, by (11) and Theorem 1(4) we have $\begin{matrix} G^{*} (β) (y) \\ \geq F (α_{0}) (x_{0}) \otimes (F (α_{0}) (x_{0}) \to_{\otimes} G (β) (y)) \\ = 1 \otimes (1 \to_{\otimes} G (β) (y)) = G (β) (y) . \end{matrix}$ On the other hand, since F (α) (x) → _⊗G (β) (y) ≤ F (α) (x) → _⊗G (β) (y) is always true, it follows from Theorem 1(7) that F (α) (x) ⊗ (F (α) (x) → _⊗G (β) (y)) ≤ G (β) (y) and consequently

$G^{*} (β) (y) = \sup_{α \in A} \sup_{x \in U} F^{*} (α) (x) \otimes (F (α) (x) \to_{\otimes} G (β) (y)) \leq G (β) (y)$ .

This proves that G^∗ (β) = G (β) and hence (G^∗, B) = (G, B).

Now we consider fuzzy soft modus tollens (briefly, FSMT):

Major premise:	(F, A) → (G, B)
Minor premise:	(G^∗, B) (15)
Conclusion:	(F^∗, A)

where (F, A), (F^∗, A) and (G, B), (G^∗, B) are fuzzy soft sets over U and V, respectively.

Principle of triple I for FSMT: The FSMT conclusion (F^∗, A) of (15) is the largest fuzzy soft set over U satisfying (10) for each α ∈ A, β ∈ B, x ∈ U, y ∈ V.

Theorem 4. Suppose that ⊗ is a left-continuoust-norm and the implication operator in FSMT is →_⊗, then the Triple I solution of (15) can be expressed as follows: for each α ∈ A, x ∈ U,

$\begin{matrix} F^{*} (α) (x) & = & \inf_{β \in B} \inf_{y \in V} ((F (α) (x) \\ \to_{\otimes} G (β) (y)) \to_{\otimes} G^{*} (β) (y)) . \end{matrix}$ (16)

Proof. First, we prove that (F^∗, A) determined by (16) satisfies (10). For any β ∈ B, y ∈ V, α ∈ A, x ∈ U, it follows from (16) that $\begin{matrix} F^{*} (α) (x) \\ \leq ((F (α) (x) \to_{\otimes} G (β) (y)) \to_{\otimes} G^{*} (β) (y)), \\ (F (α) (x) \to_{\otimes} G (β) (y)) \\ \to_{\otimes} (F^{*} (α) (x) \to_{\otimes} G^{*} (β) (y)) \\ = F^{*} (α) (x) \to_{\otimes} ((F (α) (x) \to_{\otimes} G (β) (y)) \\ \to_{\otimes} G^{*} (β) (y)) = 1 . \end{matrix}$ Next, we will verify that (F^∗, A) is the largest fuzzy soft set over U satisfying (10). Assume that the fuzzy soft set (H, A) satisfies (10), that is $\begin{matrix} (F (α) (x) \to_{\otimes} G (β) (y)) \\ \to_{\otimes} (H (α) (x) \to_{\otimes} G^{*} (β) (y)) = 1 \end{matrix}$ for any β ∈ B, y ∈ V, α ∈ A, x ∈ U. It follows that $\begin{matrix} H (α) (x) & \to_{\otimes} ((F (α) (x) \to_{\otimes} G (β) (y)) \\ \to_{\otimes} G^{*} (β) (y)) = 1, \\ H (α) (x) & \leq (F (α) (x) \to_{\otimes} G (β) (y)) \\ \to_{\otimes} G^{*} (β) (y), \end{matrix}$ and hence $\begin{matrix} H (α) (x) & \leq \inf_{β \in B} \inf_{y \in V} ((F (α) (x) \\ \to_{\otimes} G (β) (y)) \to_{\otimes} G^{*} (β) (y)) = F^{*} (α) (x) . \end{matrix}$ Consequently, (H, A) ⊆ (F^∗, A) as required.

Corollary 4. Suppose that the implication operator in FSMT is →_{⊗_L}, then the Triple I solution of (15) can be expressed as follows: for each α ∈ A, x ∈ U,

$\begin{matrix} F^{*} (α) (x) & = & \inf_{β \in B} \inf_{y \in E_{(α, β, x)}} (F (α) (x) \\ - G (β) (y) + G^{*} (β) (y)) \lor G^{*} (β) (y) \end{matrix}$ (17)

where E_(α,β,x) = {y ∈ V ; F (α) (x) - G (β) (y) + G^∗ (β) (x) <1}.

Proof. If y ∈ E_(α,β,x), then F (α) (x) - G (β) (y) + G^∗ (β) (x) <1 and hence $\begin{matrix} (F (α) (x) \to_{\otimes_{L}} G (β) (y)) \to_{\otimes_{L}} G^{*} (β) (y) \\ = (1 - (1 - F (α) (x) + G (β) (y)) \land 1 \\ + G^{*} (β) (y)) \land 1 \\ = ((F (α) (x) - G (β) (y) + G^{*} (β) (y)) \\ \lor G^{*} (β) (y)) \land 1 \\ = (F (α) (x) - G (β) (y) + G^{*} (β) (y)) \lor G^{*} (β) (y) . \end{matrix}$ If y ∉ E_(α,β,x), then F (α) (x) - G (β) (y) + G^∗ (β) (x) ≥1 and hence $\begin{matrix} (F (α) (x) \to_{\otimes_{L}} G (β) (y)) \to_{\otimes_{L}} G^{*} (β) (y) \\ = ((F (α) (x) - G (β) (y) + G^{*} (β) (y)) \\ \lor G^{*} (β) (y)) \land 1 \\ = (F (α) (x) - G (β) (y) + G^{*} (β) (y)) \land 1 = 1 . \end{matrix}$ It follows from (16) that

$F^{*} (α) (x) = \inf_{β \in B} \inf_{y \in E_{(α, β, x)}} (F (α) (x) - G (β) (y) + G^{*} (β) (y)) \lor G^{*} (β) (y)$ .

Similarly, we have

Corollary 5. Suppose that the implication operator in FSMT is →_⊗₀, then the Triple I solution of (15) can be expressed as follows: for each α ∈ A, x ∈ U,

$\begin{matrix} F^{*} (α) (x) = \inf_{β \in B} \inf_{y \in E_{(α, β, x)}} \\ (1 - (F (α) (x) \to_{\otimes_{0}} G (β) (y))) \lor G^{*} (β) (y) \end{matrix}$ (18)

where E_(α,β,x) = {y ∈ V ; F (α) (x) → _⊗₀G (β) (y) > G^∗ (β) (x)}.

Let → be a fuzzy implication operator. The negation n (x) induced by → is defined as n (x) = x → 0 for each x ∈ [0, 1]. n (x) is called involution negation [5] if n (n (x)) = x. Clearly, the negation induced by → is an involution negation if (x → 0) →0 = x. It is trivial to verify that the negations induced by →_{⊗_L} and →_⊗₀ are involution negations, whereas the negations induced by →_{⊗_G} and →_{⊗_π} are not involution negations.

Theorem 5. Suppose that ⊗ is a left-continuoust-norm, the implication operator in FSMT is →_⊗, the negation induced by →_⊗ is an involution negation, and there exist β ∈ B, y ∈ V such that G (β) (y) =0. Then the Triple I method for FSMT is reversible, i.e., if the fuzzy soft set (G^∗, B) in (15) equals (G, B), then (F^∗, A) equals (F, A).

Proof. Suppose that (G^∗, B) = (G, B) in (15), and β₀ ∈ B, y₀ ∈ V such that G (β₀) (y₀) =0. For each α ∈ A and x ∈ U, by (16) we have

\begin{matrix} F^{*} (α) (x) \\ \leq (F (α) (x) \to_{\otimes} G (β_{0}) (y_{0})) \to_{\otimes} G (β_{0}) (y_{0}) \\ = (F (α) (x) \to_{\otimes} 0) \to_{\otimes} 0 = F (α) (x) . \end{matrix}

On the other hand, we have F (α) (x) ≤ (F (α) (x) → _⊗G (β) (y)) → _⊗G (β) (y) and consequently F (α) (x) ≤ ∧ _β∈B ∧ _y∈V ((F (α) (x) → _⊗G (β) (y)) → _⊗G (β) (y)) = F^∗ (α) (x) . This proves that F^∗ (α) = F (α) and hence (F^∗, A) = (F, A).

5 An application of FSMP in medical diagnosis

In this section, we present a medical diagnosis problem where FSMP model can be applied. According to [22], Dengue has emerged as a global problem since the Second World War and is endemic in several countries. The symptoms range from mild fever, to incapacitating high fever, with severe headache, pain behind the eyes, muscle and joint pain. There is no vaccine or any precise medicine to treat dengue. Early recognition and prompt supportive treatment can substantially lower the risk of medical complications and death. Muthukumar [22] presented an similarity measure based approach to estimate the possibility that a patient having certain visible symptoms is suffering from dengue fever: Firstly, the typical model IFSS (intuitionistic fuzzy soft set) for dengue fever and the IFSSs of symptoms for the patients are constructed. Next, the similarity measures for these IFSSs with the typical model IFSS are computed. If the value of similarity measure is greater than a given threshold value, then it is concluded that the person is possibly suffering from dengue fever. Now, we discuss this problem by approximate reasoning.

Let E = {e₁, e₂, e₃} be a set of symptoms, where e₁, e₂ and e₃ stand for ‘sudden fever’, ‘headache’ and ‘knee pains’ respectively. Further, we use linguistic labels ‘no’, ‘mild’ and ‘severe’, denoted by NO, MD and SV respectively, to describe the state of medical indicators. A typical model for dengue fever can be prepared with the help of a physician and be presented by a fuzzy soft set (F, E) on the universe X, where X = {NO, MD, SV}. Suppose that: $\begin{matrix} F (e_{1}) = 0.1 / NO + 0.3 / MD + 0.6 / SV; \\ F (e_{2}) = 0.2 / NO + 0.5 / MD + 0.3 / SV; \\ F (e_{3}) = 0.1 / NO + 0.4 / MD + 0.5 / SV . \end{matrix}$ Further, we suppose that a patient with the typical symptoms described by (F, E) is suffering from dengue fever in grade 0.9, and is not suffering from dengue fever in grade 0.1. Thus we can form a fuzzy soft inference rule (F, E) → (G, B), where B = {β (Dengue)}, and G (β) =0.1/NO + 0.9/YES.

Let Mr.Z be a patient with some symptoms related to dengue fever. The states of medical indicators of Mr.Z are presented as a fuzzy soft set (F^∗, E), where $\begin{matrix} F^{*} (e_{1}) = 0.3 / NO + 0.1 / MD + 0.6 / SV; \\ F^{*} (e_{2}) = 0.5 / NO + 0.2 / MD + 0.3 / SV; \\ F^{*} (e_{3}) = 0.1 / NO + 0.5 / MD + 0.4 / SV . \end{matrix}$

By triple I method for FSMP, we can present an inference result given by a fuzzy soft set (G^∗, B). For simplicity, we use →_{⊗_G} as the fuzzy implication operator. According to Corollary 2, we have $\begin{matrix} sup_{x \in X} F^{*} (e_{1}) (x) \land (F (e_{1}) (x) \to_{\otimes_{G}} G (β) (NO)) \\ = (0.3 \land (0.1 \to_{\otimes_{G}} 0.1)) \lor (0.1 \land (0.3 \to_{\otimes_{G}} 0.1)) \\ \lor (0.6 \land (0.6 \to_{\otimes_{G}} 0.1)) = 0.3, \end{matrix}$ $\begin{matrix} \sup_{x \in X} F^{*} (e_{2}) (x) \land (F (e_{2}) (x) \to_{\otimes_{G}} G (β) (NO)) \\ = (0.5 \land (0.2 \to_{\otimes_{G}} 0.1)) \lor (0.2 \land (0.5 \to_{\otimes_{G}} 0.1)) \\ \lor (0.3 \land (0.3 \to_{\otimes_{G}} 0.1)) = 0.1, \end{matrix}$ $\begin{matrix} \sup_{x \in X} F^{*} (e_{3}) (x) \land (F (e_{3}) (x) \to_{\otimes_{G}} G (β) (NO)) \\ = (0.1 \land (0.1 \to_{\otimes_{G}} 0.1)) \lor (0.5 \land (0.4 \to_{\otimes_{G}} 0.1)) \\ \lor (0.4 \land (0.5 \to_{\otimes_{G}} 0.1)) = 0.1 . \end{matrix}$ Thus, $G^{*} (β) (NO) = \sup_{α \in E} \sup_{x \in X} F^{*} (α) (x) \land (F (α) (x) \to_{\otimes_{G}} G (β) (NO)) = 0.3$ . Similarly, by routine computation, we have G^∗ (β) (YES) =0.6. Thus we can conclude that: There is 60 percent chance that Mr.Z is suffering from dengue fever, and there is 30 percent chance that Mr.Z is not suffering from dengue fever.

This is only a simple model to show the possibility of using fuzzy soft approximate reasoning for diagnosis of disease which could be improved by incorporating clinical results and other competing diagnosis.

6 Concluding remarks

Soft set theory was originally proposed as a general mathematical tool for dealing with uncertainties. In this paper, a novel fuzzy soft set based approximate reasoning approach is presented. The notion of fuzzy soft implication relation between two fuzzy soft sets is introduced and the composition method of fuzzy soft implication relations is proposed. Furthermore, The Triple I methods for fuzzy soft modus ponens (FSMP)and fuzzy soft modus tollens (FSMT) with respect to left-continuous t-norms and its residual implication are presented and the reversibility properties of Triple I methods are analyzed.

In further research, The Triple I method for FSMP and FSMT with respect to diverse implication operators and their relationships are important and interesting issues to be addressed. Moreover, the application of fuzzy soft reasoning in real life situations need to be deeply discussed.

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