Similarity measure for soft matrices and its applications

Abstract

In this paper, the symmetric difference operation for each of the soft sets and the soft matrices is defined and then their related properties are derived. Also, the concept of similarity measure for the soft matrices is introduced. Thus, we pioneer the budding phase of the use of soft matrices to find the similarity of soft sets. Through the soft matrices, we develop the Scilab code which facilitates the computation process of similarity measure among the soft sets composed of many parameters and alternatives. Moreover, it is shown with several examples that the similarity measure can be used for some cases involving the uncertainties in various areas. To confirm the consistency of the emerging similarity measure, we make comparisons with the preexisting similarity measures for the soft sets.

Keywords

Soft set soft matrix symmetric difference similarity measure significantly similar

1 Introduction

Many researchers who make investigation in the fields such as engineering, economics, medical science, environmental science, sociology often encounter the complexity of uncertain or unknown data. The frame of uncertainty appearing in these fields may be very different. Therefore, according to the frame of uncertainty, many mathematical tools such as the fuzzy sets, intuitionistic fuzzy sets, interval-valued fuzzy sets, neutrosophic sets and rough sets were developed to describe and overcome the uncertainties. Soft set theory was proposed by Molodtsov [16] as a novel mathematical tool to deal with structures involving uncertain or unknown data. In recent years, the researches on soft set theory have been active, and so great progress has been achieved. Maji et al. [13] presented a work which was an introduction to the operations on the soft sets. They defined the subset and complement of a soft set, the soft union, the soft intersection and the soft binary operations. Pei and Miao [17] gave an alternative definition for the soft intersection. Ali et al. [1] introduced the restricted intersection, the extended intersection, the restricted union and the restricted difference of two soft sets, and also improved the complement of a soft set in [13]. Sezgin and Atagün [18] published a study extending the theoretical aspects of soft operations. Moreover, they described the concept of restricted symmetric difference of soft sets. Zhu and Wen [23] depicted some of the operations on the soft sets with a new perspective. Çagman and Enginoglu [8] redefined the union, intersection, difference of soft sets to make them more useful in some cases. In [2 , 19], the soft operations were derived for the soft set family. The researchers in [4 , 21] studied on the relations and mappings of soft structures. In addition to these, Maji et al. [14] depicted the soft sets in the unified format of a binary information table. After this depiction, the soft matrices which are symbolically equivalent to soft sets were created in [7]. Also, some basic operations on these matrices were investigated in detail. Later, some products such as the generalized products and the row-products on the soft matrices were discussed in [3, 10]. Basu et al. [5] developed the addition and subtraction of soft matrices and their some properties. Meanwhile, Kamacı et al. [11] derived the operations of difference, restricted difference, extended difference and weak-extended difference for the soft sets and the soft matrices. The similarity is a topic, which is frequently discussed in various scientific fields, and has many advantages. Lately, as in other mathematical tools, the research on the similarity of soft sets is very popular. Majumdar and Samanta [15] published an initial work on the similarity measure of soft sets. Immediately afterwards, Kaharal [12] introduced a new similarity measure improving the similarity measure defined in [15]. Moreover, Yang [22] presented a counterexample showing that the similarity measure introduced in [12] contains error, and then described a novel similarity measure away from this error. In [6], some properties concerning the similarity measure of soft sets were deeply investigated. In this study, our goal is to present new soft operations and a novel similarity measure achieving the similarity of two soft sets whatever their nature over a common universal set. Furthermore, it is to introduce the similarity measure for the soft matrices which make the operations more functional in obtaining the results. The remainder of this paper is organized as below. Section 2 is dedicated to some basic concepts and operations on the soft sets and the soft matrices. Section 3 introduces the symmetric difference of two soft sets and its properties. In Section 4, the similarity measure between two soft sets is newly developed and several results are presented. In Section 5, the symmetric difference of two soft matrices is practically defined. In Section 6, the similarity measure between two soft matrices is constructed, and then its computing code is created. Section 7 is devoted to some applications of the proposed similarity measure and comparison of it with the existing similarity measures. The final section contains the conclusions.

2 Preliminaries

In this section, the definition of soft set and its operations are given. Later on, the soft matrix corresponding to the soft set is recalled. Also, some operations of the soft matrices are presented.

From now on, U = {u₁, u₂,…, u_n} is a universal set, P (U) is the power set of U, E = {e₁, e₂,…, e_m} is a set of parameters and A ⊆ E.

Definition 2.1. ([8, 16]) The set of ordered pairs F_A = {(e_j, f_A (e_j)): e_j ∈ E, f_A (e_j) ∈ P (U)}

is called a soft set over U, where f_A: E → P (U) such that f_A (e_j) =∅ if e_j ∉ A. Also, f_A is said to be an approximate function of the soft set F_A.

Note: The set of all soft sets over U will be denoted by S (U).

Example 2.2. Let U = {u₁, u₂, u₃, u₄, u₅} be a universal set and E = {e₁, e₂, e₃, e₄, e₅, e₆} be a set of parameters. If A = {e₁, e₃, e₄, e₅} ⊂ E and f_A: E → P (U) such that f_A (e₁) = {u₁, u₃}, f_A (e₃) = {u₂, u₃, u₅}, f_A (e₄) =∅, f_A (e₅) = U, then we obtain the following soft set. F_A = {(e₁, {u₁, u₃}), (e₃, {u₂, u₃, u₅}), (e₅, U)}.

Remark:f_A (e_j) =∅ means that there is no element in U related to the parameter e_j ∈ E. Therefore, these elements in the soft sets are not displayed, as it is pointless to consider such parameters.

Definition 2.3. ([8]) Let F_A ∈ S (U). Then, F_A is called [(a)] an empty soft set if f_A (e_j) =∅ for all e_j ∈ E and it is denoted by F_∅. [(b)] a universal soft set if f_A (e_j) = U for all e_j ∈ E and it is denoted by $F_{\tilde{E}}$ .

Definition 2.4. ([8]) Let F_A, F_B ∈ S (U). Then [(a)] F_A is a soft subset of F_B if f_A (e_j) ⊆ f_B (e_j) for all e_j ∈ E and it is denoted by $F_{A} \tilde{\subseteq} F_{B}$ . [(b)] F_A and F_B are soft equal if f_A (e_j) = f_B (e_j) for all e_j ∈ E and it is denoted by F_A = F_B.

Definition 2.5. ([8]) Let F_A, F_B ∈ S (U). Then, the soft set F_C is called [(a)] complement of F_A if f_C (e_j) = U \ f_A (e_j) for all e_j ∈ E and it is denoted by $F_{A}^{c}$ . [(b)] union of F_A and F_B if f_C (e_j) = f_A (e_j) ∪ f_B (e_j) for all e_j ∈ E and it is denoted by $F_{A} \tilde{\cup} F_{B}$ . [(c)] intersection of F_A and F_B if f_C (e_j) = f_A (e_j) ∩ f_B (e_j) for all e_j ∈ E and it is denoted by $F_{A} \tilde{\cap} F_{B}$ .

In the following, the representation in the form of binary information table of soft set is described and then the soft matrix constructed by using this representation is introduced.

Definition 2.6. ([7]) Let F_A ∈ S (U). Then, a subset of U × E is uniquely defined by R_A = {(u_i, e_j): e_j ∈ E and u_i ∈ f_A (e_j)}.

which is said to be a relation form of F_A. The characteristic function of R_A is written as χ_{R
_A}: U × E → {0, 1} such that $χ_{R_{A}} (u_{i}, e_{j}) = {\begin{matrix} 1, & if (u_{i}, e_{j}) \in R_{A} \\ 0, & if (u_{i}, e_{j}) \notin R_{A} \end{matrix}$

Then, R_A can be given by a binary table as in the following form: If a_ij = χ_{R
_A} (u_i, e_j), we can define a matrix $[a_{ij}]_{n \times m} = [\begin{matrix} a_{11} & a_{12} & . & . & . & a_{1 m} \\ a_{21} & a_{22} & . & . & . & a_{2 m} \\ . & . & . & . \\ . & . & . & . \\ . & . & . & . \\ a_{n 1} & a_{n 2} & . & . & . & a_{nm} \end{matrix}]$

which is called a soft matrix corresponding to the soft set F_A over U.

Note: The set of all soft matrices corresponding to the soft sets in S (U) will be denoted by SM (U).

Example 2.7. Consider the soft set F_A in Example 2.2. Then, the relation form of F_A is R_A = {(u₁, e₁), (u₁, e₅), (u₂, e₃), (u₂, e₅), (u₃, e₁), (u₃, e₃), (u₃, e₅), (u₄, e₅), (u₅, e₃), (u₅, e₅)}. Thus, we obtain soft matrix $[a_{ij}]_{5 \times 6} = [\begin{matrix} 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 \end{matrix}] .$

Definition 2.8. ([7]) Let [a_ij] ∈ SM (U). Then [a_ij] is called [(a)] a zero soft matrix if a_ij = 0 for all i, j and it is denoted by [0]. [(b)] a universal soft matrix if a_ij = 1 for all i, j and it is denoted by [1].

Definition 2.9. ([7]) Let [a_ij], [b_ij] ∈ SM (U). Then [(a)] [a_ij] is a soft submatrix of [b_ij] if a_ij ≤ b_ij for all i, j and it is denoted by $[a_{ij}] \tilde{\subseteq} [b_{ij}]$ . [(b)] [a_ij] and [b_ij] are soft equal matrices if a_ij = b_ij for all i, j and it is denoted by [a_ij] = [b_ij].

Definition 2.10. ([7]) Let [a_ij], [b_ij] ∈ SM (U). Then, the soft matrix [c_ij] is called [(a)] complement of [a_ij] if c_ij = 1 - a_ij for all i, j and it is denoted by [a_ij] ^c. [(b)] union of [a_ij] and [b_ij] if c_ij = max {a_ij, b_ij} for all i, j and it is denoted by $[a_{ij}] \tilde{\cup} [b_{ij}]$ . [(c)] intersection of [a_ij] and [b_ij] if c_ij = min {a_ij, b_ij} for all i, j and it is denoted by $[a_{ij}] \tilde{\cap} [b_{ij}]$ .

3 Symmetric difference of soft sets

In this section, new properties of the difference of soft sets are investigated. Also, the concept of symmetric difference of soft sets is introduced and then its operations are derived.

Definition 3.1. ([8]) Let F_A, F_B ∈ S (U). Then, the difference of F_A and F_B is a soft set defined by the approximate function $f_{A \tilde{\} B} (e_{j}) = f_{A} (e_{j}) \ f_{B} (e_{j})$ for all e_j ∈ E and it is denoted by $F_{A} \tilde{\} F_{B}$ .

Example 3.2. Let U = {u₁, u₂, u₃, u₄} be a universal set, E = {e₁, e₂, e₃, e₄, e₅, e₆, e₇, e₈, e₉} be a set of parameters. Also, let’s consider A = {e₂, e₃, e₄, e₆, e₈} and B = {e₁, e₃, e₄, e₆, e₈, e₉}. For these subset of parameter set, we take the following soft sets.

F_A = {(e₂, {u₁, u₂}), (e₃, {u₁, u₂, u₃}), (e₄, {u₂, u₃, u₄}), (e₆, {u₂})},F_B = {(e₁, {u₂, u₃}), (e₃, {u₂, u₄}), (e₄, U), (e₆, {u₃, u₄}), (e₉, {u₁, u₄})}. Then, we obtain the difference of F_A and F_B as follows $F_{A} \tilde{\} F_{B} = {(e_{2}, {u_{1}, u_{2}}), (e_{3}, {u_{1}, u_{3}}), (e_{6} {u_{2}})} .$

Proposition 3.3. ([8]) Let F_A, F_B, F_C ∈ S (U). [(i)] $F_{A} \tilde{\} F_{B} = F_{A} \tilde{\cap} F_{B}^{c}$ . [(ii)] $F_{A} \tilde{\} F_{B} = F_{\emptyset} \Leftrightarrow F_{A} \tilde{\subseteq} F_{B}$ . [(iii)] A∩ B = ∅ ⇒ $F_{A} \tilde{\} F_{B} = F_{A}$ and $F_{B} \tilde{\} F_{A} = F_{B}$ .

Now, we derive new properties for the difference of soft sets.

Proposition 3.4. Let F_A, F_B, F_C ∈ S (U). [(i)] $F_{A} \tilde{\} F_{A} = F_{\emptyset}$ . [(ii)] $F_{A} \tilde{\} F_{A}^{c} = F_{A}$ . [(iii)] $F_{A} \tilde{\} F_{\emptyset} = F_{A}$ and $F_{\emptyset} \tilde{\} F_{A} = F_{\emptyset}$ . [(iv)] $F_{\tilde{E}} \tilde{\} F_{A} = F_{A}^{c}$ and $F_{A} \tilde{\} F_{\tilde{E}} = F_{\emptyset}$ . [(v)] $(F_{A} \tilde{\cap} F_{B}) \tilde{\} F_{C} = (F_{A} \tilde{\} F_{C}) \tilde{\cap} (F_{B} \tilde{\} F_{C})$ . [(vi)] $(F_{A} \tilde{\cup} F_{B}) \tilde{\} F_{C} = (F_{A} \tilde{\} F_{C}) \tilde{\cup} (F_{B} \tilde{\} F_{C})$ . [(vii)] $(F_{A} \tilde{\} F_{B}) \tilde{\} F_{C} = (F_{A} \tilde{\} F_{C}) \tilde{\} (F_{B} \tilde{\} F_{C})$ . [(viii)] $F_{A} \tilde{\} (F_{B} \tilde{\cap} F_{C}) = (F_{A} \tilde{\} F_{B}) \tilde{\cup} (F_{A} \tilde{\} F_{C})$ . [(ix)] $F_{A} \tilde{\} (F_{B} \tilde{\cup} F_{C}) = (F_{A} \tilde{\} F_{B}) \tilde{\cap} (F_{A} \tilde{\} F_{C})$ . [(x)] $(F_{A} \tilde{\} F_{B}) \tilde{\cap} F_{C} = (F_{A} \tilde{\cap} F_{C}) \tilde{\} (F_{B} \tilde{\cap} F_{C})$ . [(xi)] $F_{A} \tilde{\cap} (F_{B} \tilde{\} F_{C}) = (F_{A} \tilde{\cap} F_{B}) \tilde{\} (F_{A} \tilde{\cap} F_{C})$ .

Proof. Let F_A, F_B, F_C ∈ S (U).The proofs of (i)-(iv) are straightforward, therefore omitted.(v) Assume that $F_{A} \tilde{\cap} F_{B} = F_{D}$ where f_D (e_j) = f_A (e_j) ∩ f_B (e_j) for all e_j ∈ E. And suppose that $F_{D} \tilde{\} F_{C} = F_{G}$ where f_G (e_j) = f_D (e_j) \ f_C (e_j) = (f_A (e_j) ∩ f_B (e_j)) \ f_C (e_j) = (f_A (e_j) \ f_C (e_j)) ∩ (f_B (e_j) \ f_C (e_j)) for all e_j ∈ E.Now, we deal with the other side of equality. Assume that $F_{A} \tilde{\} F_{C} = F_{H}$ where f_H (e_j) = f_A (e_j) \ f_C (e_j) for all e_j ∈ E, and also $F_{B} \tilde{\} F_{C} = F_{I}$ where f_I (e_j) = f_B (e_j) \ f_C (e_j) for all e_j ∈ E. Suppose that $F_{H} \tilde{\cap} F_{I} = F_{L}$ where f_L (e_j) = f_H (e_j) ∩ f_I (e_j) = (f_A (e_j) \ f_C (e_j)) ∩ (f_B (e_j) \ f_C (e_j)) for all e_j ∈ E. Since f_G and f_L are the same set-valued approximate functions for all e_j ∈ E, the proof is completed.(vi)-(vii) The proofs can be proved similar to (v), hence omitted.(viii) Suppose that $F_{B} \tilde{\cap} F_{C} = F_{D}$ where f_D (e_j) = f_B (e_j) ∩ f_C (e_j) for all e_j ∈ E. And suppose that $F_{A} \tilde{\} F_{D} = F_{G}$ where f_G (e_j) = f_A (e_j) \ f_D (e_j) = f_A (e_j) \ (f_B (e_j) ∩ f_C (e_j)) = (f_A (e_j) \ f_B (e_j)) ∪ (f_A (e_j) \ f_C (e_j)) for all e_j ∈ E.Now, we deal with the other side of equality. Assume that $F_{A} \tilde{\} F_{B} = F_{H}$ where f_H (e_j) = f_A (e_j) \ f_B (e_j) for all e_j ∈ E, and also $F_{A} \tilde{\} F_{C} = F_{I}$ where f_I (e_j) = f_A (e_j) \ f_C (e_j) for all e_j ∈ E. Assume that $F_{H} \tilde{\cup} F_{I} = F_{L}$ where f_L (e_j) = f_H (e_j) ∪ f_I (e_j) = (f_A (e_j) \ f_B (e_j)) ∪ (f_A (e_j) \ f_C (e_j)) for all e_j ∈ E. Since f_G and f_L are the same set-valued approximate functions for all e_j ∈ E, the proof is completed.(ix) The proof can be proved similar to (viii), hence omitted.(x) Assume that $F_{A} \tilde{\} F_{B} = F_{D}$ where f_D (e_j) = f_A (e_j) \ f_B (e_j) for all e_j ∈ E. And suppose that $F_{D} \tilde{\cap} F_{C} = F_{G}$ where f_G (e_j) = f_D (e_j) ∩ f_C (e_j) = (f_A (e_j) \ f_B (e_j)) ∩ f_C (e_j)) = (f_A (e_j) ∩ f_C (e_j)) \ (f_B (e_j) ∩ f_C (e_j)) for all e_j ∈ E.Now, we handle the other side of equality. Let’s take $F_{A} \tilde{\cap} F_{C} = F_{H}$ where f_H (e_j) = f_A (e_j) ∩ f_C (e_j) for all e_j ∈ E, and also $F_{B} \tilde{\cap} F_{C} = F_{I}$ where f_I (e_j) = f_B (e_j) ∩ f_C (e_j) for all e_j ∈ E. Then, we can write $F_{H} \tilde{\} F_{I} = F_{L}$ where f_L (e_j) = f_H (e_j) \ f_I (e_j) = (f_A (e_j) ∩ f_C (e_j)) \ (f_B (e_j) ∩ f_C (e_j)) for all e_j ∈ E. Thus, it is obtained that f_G and f_L are the same set-valued approximate functions for all e_j ∈ E. Hence, we conclude the proof.(xi) The proof can be illustrated similar to (x), hence it is omitted.□

We now define symmetric difference of two soft sets and investigate its related properties.

Definition 3.5. Let F_A, F_B ∈ S (U). Then, the symmetric difference of F_A and F_B is a soft set defined by the approximate function $f_{A \tilde{△} B} (e_{j}) = f_{A} (e_{j}) △ f_{B} (e_{j})$

for all e_j ∈ E. It is denoted by $F_{A} \tilde{△} F_{B}$ .Here, f_A (e_j) △ f_B (e_j) = (f_A (e_j) ∪ f_B (e_j)) \ (f_A (e_j) ∩ f_B (e_j)).

Example 3.6. Let’s reconsider the soft sets F_A and F_B in Example 3.2. Then, we obtain the symmetric difference of these soft sets as follows $F_{A} \tilde{△} F_{B} = {(e_{1}, {u_{2}, u_{3}}), (e_{2}, {u_{1}, u_{2}}),$ (e₃, {u₁, u₃, u₄}), (e₄, {u₁}), (e₆, {u₂, u₃, u₄}), (e₉, {u₁, u₄})}.

Proposition 3.7. Let F_A, F_B, F_C ∈ S (U). [(i)] $F_{A} \tilde{△} F_{A} = F_{\emptyset}$ . [(ii)] $F_{A} \tilde{△} F_{A}^{c} = F_{\tilde{E}}$ . [(iii)] $F_{A} \tilde{△} F_{\emptyset} = F_{A}$ . [(iv)] $F_{\tilde{E}} \tilde{△} F_{A} = F_{A}^{c}$ . [(v)] $F_{A} \tilde{△} F_{B} = F_{B} \tilde{△} F_{A}$ . [(vi)] $F_{A} \tilde{△} F_{B} = (F_{A} \tilde{\} F_{B}) \tilde{\cap} (F_{B} \tilde{\} F_{A})$ . [(vii)] $F_{A} \tilde{△} F_{B} = F_{A}^{c} \tilde{△} F_{B}^{c}$ . [(viii)] $(F_{A} \tilde{△} F_{B}) \tilde{△} F_{C} = F_{A} \tilde{△} (F_{B} \tilde{△} F_{C})$ .

proof. Let F_A, F_B, F_C ∈ S (U).The proofs of (i)-(v) can be simply illustrated, therefore omitted.(vi) Suppose that $F_{A} \tilde{△} F_{B} = F_{D}$ where $\begin{array}{l} f_{D} (e_{j}) = f_{A} (e_{j}) △ f_{B} (e_{j}) = \\ (f_{A} (e_{j}) \cup f_{B} (e_{j})) ∖ (f_{A} (e_{j}) \cap f_{B} (e_{j})) \\ = (f_{A} (e_{j}) \cup f_{B} (e_{j})) \cap {(f_{A} (e_{j}) \cap f_{B} (e_{j}))}^{c} \\ = (f_{A} (e_{j}) \cup f_{B} (e_{j})) \cap (f_{A}^{c} (e_{j}) \cup f_{B}^{c} (e_{j})) \\ = [(f_{A} (e_{j}) \cup f_{B} (e_{j})) \cap f_{A}^{c} (e_{j})] \cup [(f_{A} (e_{j}) \cup f_{B} (e_{j})) \cap f_{B}^{c} (e_{j})] \\ = [(f_{A} (e_{j}) \cap f_{A}^{c} (e_{j})) \cup (f_{B} (e_{j}) \cap f_{A}^{c} (e_{j}))] \cup [(f_{A} (e_{j}) \cap f_{B}^{c} (e_{j})) \cup (f_{B} (e_{j}) \cap f_{B}^{c} (e_{j}))] \\ = (f_{B} (e_{j}) \cap f_{A}^{c} (e_{j})) \cup (f_{A} (e_{j}) \cap f_{B}^{c} (e_{j})) \\ = (f_{B} (e_{j}) ∖ f_{A} (e_{j})) \cup (f_{A} (e_{j}) ∖ f_{B} (e_{j})) \end{array}$ for all e_j ∈ E. Now, we let’s take $F_{A} \tilde{\} F_{B} = F_{G}$ where f_G (e_j) = f_A (e_j) \ f_B (e_j) for all e_j ∈ E, and also $F_{B} \tilde{\} F_{A} = F_{H}$ where f_H (e_j) = f_B (e_j) \ f_A (e_j) for all e_j ∈ E. Assume that $F_{G} \tilde{\cup} F_{H} = F_{I}$ where f_I (e_j) = f_G (e_j) ∪ f_H (e_j) = (f_A (e_j) \ f_B (e_j)) ∪ (f_B (e_j) \ f_A (e_j)) for all e_j ∈ E. This means that f_D and f_I are the same set-valued approximate functions for all e_j ∈ E. Hence, we complete the proof.(vii) Suppose that $F_{A} \tilde{△} F_{B} = F_{D}$ where f_D (e_j) = f_A (e_j) △ f_B (e_j) = (f_A (e_j) \ f_B (e_j)) ∪ (f_B (e_j) \ f_A (e_j)) $= (f_{A} (e_{j}) \cap f_{B}^{c} (e_{j})) \cup (f_{B} (e_{j}) \cap f_{A}^{c} (e_{j}))$ $= (f_{A}^{c} (e_{j}) \cap f_{B} (e_{j})) \cup (f_{B}^{c} (e_{j}) \cap f_{A} (e_{j}))$ $= (f_{A}^{c} (e_{j}) \ f_{B}^{c} (e_{j})) \cup (f_{B}^{c} (e_{j}) \ f_{A}^{c} (e_{j}))$ for all e_j ∈ E. Now, we consider $F_{A}^{c} \tilde{\} F_{B}^{c} = F_{G}$ where $f_{G} (e_{j}) = f_{A}^{c} (e_{j}) △ f_{B}^{c} (e_{j}) = (f_{A}^{c} (e_{j}) \ f_{B}^{c} (e_{j})) \cup (f_{B}^{c} (e_{j}) \ f_{A}^{c} (e_{j}))$ for all e_j ∈ E. This follows that f_D and f_G are the same set-valued approximate functions for all e_j ∈ E. Thus, the proof is completed.(viii) Suppose that $F_{A} \tilde{△} F_{B} = F_{D}$ where $f_{D} (e_{j}) = f_{A} (e_{j}) △ f_{B} (e_{j}) = (f_{A} (e_{j}) \cap f_{B}^{c} (e_{j})) \cup (f_{A}^{c} (e_{j}) \cap f_{B} (e_{j}))$ for all e_j ∈ E. And assume that $F_{D} \tilde{△} F_{C} = F_{G}$ where f_G (e_j) = f_D (e_j) △ f_C (e_j) $= [(f_{A} (e_{j}) \cap f_{B}^{c} (e_{j})) \cup (f_{A}^{c} (e_{j}) \cap f_{B} (e_{j}))] △ f_{C} (e_{j})$ $= [[(f_{A} (e_{j}) \cap f_{B}^{c} (e_{j})) \cup (f_{A}^{c} (e_{j}) \cap f_{B} (e_{j}))] \cap f_{C}^{c} (e_{j})] \cup [[(f_{A} (e_{j}) \cap f_{B}^{c} (e_{j})) \cup (f_{A}^{c} (e_{j}) \cap f_{B} (e_{j}))]^{c} \cap f_{C} (e_{j})]$ $= [[(f_{A} (e_{j}) \cap f_{B}^{c} (e_{j})) \cup (f_{A}^{c} (e_{j}) \cap f_{B} (e_{j}))] \cap f_{C}^{c} (e_{j})] \cup [[(f_{A}^{c} (e_{j}) \cup f_{B} (e_{j})) \cap (f_{A} (e_{j}) \cup f_{B}^{c} (e_{j}))] \cap f_{C} (e_{j})]$ $= [(f_{A} (e_{j}) \cap f_{B}^{c} (e_{j}) \cap f_{C}^{c} (e_{j})) \cup (f_{A}^{c} (e_{j}) \cap f_{B} (e_{j}) \cap f_{C}^{c} (e_{j}))] \cup [[(f_{A}^{c} (e_{j}) \cup f_{B} (e_{j})) \cap (f_{A} (e_{j}) \cup f_{B}^{c} (e_{j}))]$ ∩f_C (e_j)] $= [(f_{A} (e_{j}) \cap f_{B}^{c} (e_{j}) \cap f_{C}^{c} (e_{j})) \cup (f_{A}^{c} (e_{j}) \cap f_{B} (e_{j}) \cap f_{C}^{c} (e_{j}))] \cup [(f_{A}^{c} (e_{j}) \cup f_{B} (e_{j})) \cap [(f_{A} (e_{j}) \cap f_{C} (e_{j})) \cup$ $(f_{B}^{c} (e_{j}) \cap f_{C} (e_{j}))]]$ $= (f_{A} (e_{j}) \cap f_{B}^{c} (e_{j}) \cap f_{C}^{c} (e_{j})) \cup (f_{A}^{c} (e_{j}) \cap f_{B} (e_{j}) \cap f_{C}^{c} (e_{j})) \cup (f_{A} (e_{j}) \cap f_{B} (e_{j}) \cap f_{C} (e_{j})) \cup (f_{A}^{c} (e_{j}) \cap f_{B}^{c} (e_{j}) \cap f_{C} (e_{j}))$ $= [f_{A} (e_{j}) \cap (f_{B}^{c} (e_{j}) \cap f_{C}^{c} (e_{j}))] \cup [f_{A} (e_{j}) \cap (f_{B} (e_{j}) \cap f_{C} (e_{j}))] \cup [f_{A}^{c} (e_{j}) \cap (f_{B} (e_{j}) \cap f_{C}^{c} (e_{j}))] \cup [f_{A}^{c} (e_{j}) \cap (f_{B}^{c} (e_{j}) \cap f_{C} (e_{j}))]$ $= [f_{A} (e_{j}) \cap ((f_{B}^{c} (e_{j}) \cap f_{C}^{c} (e_{j})) \cup (f_{B} (e_{j}) \cap f_{C} (e_{j})))] \cup [f_{A}^{c} (e_{j}) \cap ((f_{B} (e_{j}) \cap f_{C}^{c} (e_{j})) \cup (f_{B}^{c} (e_{j}) \cap f_{C} (e_{j})))]$ $= [f_{A} (e_{j}) \cap ((f_{B} (e_{j}) \cup f_{C}^{c} (e_{j})) \cap (f_{B}^{c} (e_{j}) \cup f_{C} (e_{j})))] \cup [f_{A}^{c} (e_{j}) \cap ((f_{B} (e_{j}) \cap f_{C}^{c} (e_{j})) \cup (f_{B}^{c} (e_{j}) \cap f_{C} (e_{j})))]$ for all e_j ∈ E. Now, we consider $F_{B} \tilde{△} F_{C} = F_{H}$ where $f_{H} (e_{j}) = f_{B} (e_{j}) △ f_{C} (e_{j}) = (f_{B} (e_{j}) \cap f_{C}^{c} (e_{j})) \cup (f_{B}^{c} (e_{j}) \cap f_{C} (e_{j}))$ for all e_j ∈ E. And assume that $F_{A} \tilde{△} F_{H} = F_{I}$ where $f_{I} (e_{j}) = f_{A} (e_{j}) △ f_{H} (e_{j}) = [f_{A} (e_{j}) \cap ((f_{B}^{c} (e_{j}) \cup f_{C} (e_{j})) \cap (f_{B} (e_{j}) \cup f_{C}^{c} (e_{j})))] \cup [((f_{B} (e_{j}) \cap f_{C}^{c} (e_{j})) \cup (f_{B}^{c} (e_{j}) \cap f_{C} (e_{j}))) \cap f_{A}^{c} (e_{j})]$ for all e_j ∈ E. This means that f_G and f_I are the same set-valued approximate functions for all e_j ∈ E. Therefore, we complete the proof.□

4 Similarity measure between two soft sets

In this section, the similarity measure of two soft sets is defined. Afterwards, various theoretical examinations and investigations are presented in detail.

Note: From now on, | · | denotes the cardinality of a set.Let |E| = m, |U| = n and A, B ⊆ E.

Definition 4.1. Let F_A, F_B ∈ S (U). Then, the similarity measure between these soft sets is denoted and defined by

S (F_A, F_B)= $\frac{| A \cap B |}{| A \cup B |} (1 - \frac{1}{n | A \cup B |} \sum_{e_{j} \in E} | f_{A} (e_{j}) △ f_{B} (e_{j}) |)$ . Now, let’s give an example to test the change of the similarity measure with respect to the structure of two soft sets on the common universal set.

Example 4.2. Let U = {u₁, u₂, u₃, u₄} be a universal set and E = {e₁, e₂, e₃, e₄, e₅} be a set of parameters.

1. Let A = {e₁, e₃, e₄} and B = {e₁, e₃, e₄}. If F_A = {(e₁, {u₂}), (e₃, {u₁, u₂, u₄}), (e₄, {u₂, u₃})} and F_B = {(e₁, {u₂, u₃}), (e₃, {u₁, u₃}), (e₄, U)}, then we have

$F_{A} \tilde{△} F_{B} = {(e_{1}, {u_{3}}), (e_{3}, {u_{2}, u_{3}, u_{4}}), (e_{4}, {u_{1}, u_{4}})}$ .Hence, we obtain $S (F_{A}, F_{B}) = 1 - \frac{1}{2} = \frac{1}{2}$ .2. Let A = {e₁, e₃, e₄, e₅} and B = {e₁, e₃, e₄}. If F_A = {(e₁, {u₂}), (e₃, {u₁, u₂, u₄}), (e₄, {u₂, u₃})} and F_B = {(e₁, {u₂, u₃}), (e₃, {u₁, u₃}), (e₄, U)}, then $S (F_{A}, F_{B}) = \frac{15}{32}$ .3. Let A = {e₁, e₃, e₄, e₅} and B = {e₁, e₃, e₄}. If F_A = {(e₁, {u₂}), (e₃, {u₁, u₂, u₄}), (e₄, {u₂, u₃}), (e₅, {u₁, u₄})}, F_B = {(e₁, {u₂, u₃}), (e₃, {u₁, u₃}), (e₄, U)}, then $S (F_{A}, F_{B}) = \frac{3}{8}$ .4. Let A = {e₁, e₃, e₄, e₅} and B = {e₁, e₂, e₃, e₄}. If F_A = {(e₁, {u₂}), (e₃, {u₁, u₂, u₄}), (e₄, {u₂, u₃}), (e₅, {u₁, u₄})} and F_B = {(e₁, {u₂, u₃}), (e₂, {u₃}), (e₃, {u₁, u₃}), (e₄, U)}, then $S (F_{A}, F_{B}) = \frac{33}{100}$ .5. Let A = E and B = E. If F_A = {(e₁, {u₂}), (e₃, {u₁, u₂, u₄}), (e₄, {u₂, u₃}), (e₅, {u₁, u₄})} and F_B = {(e₁, {u₂, u₃}), (e₂, {u₃}), (e₃, {u₁, u₃}), (e₄, U)}, then $S (F_{A}, F_{B}) = \frac{11}{20}$ .

Proposition 4.3. Let F_A, F_B ∈ S (U). Then, we have [(i)] S (F_A, F_A) =1. [(ii)] $S (F_{A}, F_{A}^{c}) = 0$ . [(iii)] S (F_A, F_B) = S (F_B, F_A). [(iv)] 0 ≤ S (F_A, F_B) ≤1.

Proof. Let F_A, F_B ∈ S (U).(i) By Proposition 3.7 (i), we know that $F_{A} \tilde{△} F_{A} = F_{\emptyset}$ , that is, f_A (e_j)△ f_A (e_j) = ∅ for all e_j ∈ E. Hence, it is obtained that S (F_A, F_A) =1 since ∑_{e_j∈E}|f_A (e_j) △ f_A (e_j) |=0.(ii) By Proposition 3.7 (ii), we have $F_{A} \tilde{△} F_{A}^{c} = F_{\tilde{E}}$ , that is, $f_{A} (e_{j}) △ f_{A}^{c} (e_{j}) = U$ for all e_j ∈ E. Therefore, it is calculated as S (F_A, F_A) =1 since ∑_{e_j∈E}|f_A (e_j) △ f_B (e_j) | = nm.(iii) It is obvious by Proposition 3.7 (v), therefore omitted.(iv) Let’s consider γ = ∑_{e_j∈E}|f_A (e_j) △ f_B (e_j) |. Then, it must be 0 ≤ γ ≤ n|A ∪ B| since f_A (e_j)△ f_B (e_j) = ∅ for all e_j ∈ E \ (A ∪ B) from Definition 3.5. If γ = 0, then we obtain $S (F_{A}, F_{B}) = \frac{| A \cap B |}{| A \cup B |}$ . Thus, it is seen that S (F_A, F_B) ≤1. On the other hand, we compute as S (F_A, F_B) =0 for γ = n|A ∪ B|. These prove that 0 ≤ S (F_A, F_B) ≤1.□

Proposition 4.4. Let F_A, F_B ∈ S (U). If $F_{A} \tilde{\subseteq} F_{B} \tilde{\subseteq} F_{C}$ such that A ⊆ B ⊆ C ⊆ E, then S (F_A, F_C) ≤ S (F_B, F_C).

Proof. Let $F_{A} \tilde{\subseteq} F_{B} \tilde{\subseteq} F_{C}$ for A ⊆ B ⊆ C ⊆ E. Also, let γ₁ = ∑_{e_j∈E}|f_A (e_j) △ f_C (e_j) | and γ₂ = ∑_{e_j∈E}|f_B (e_j) △ f_C (e_j) |. From Definition 3.5, we can say that γ₁ ≥ γ₂. Then, we have $S (F_{A}, F_{C}) = \frac{| A |}{| C |} (1 - \frac{γ_{1}}{n | C |}) \leq \frac{| B |}{| C |} (1 - \frac{γ_{2}}{n | C |}) = S (F_{B}, F_{C})$ . Therefore, the proof is completed.□

Definition 4.5. Let F_A and F_B be two soft sets over U. Then, F_A and F_B are said to be ɛ-similar soft sets if and only if S (F_A, F_B) ≥ ɛ for ɛ ∈ (0, 1). These two soft sets are said to be significantly similar if $S (F_{A}, F_{B}) > \frac{1}{2}$ .

Example 4.6. Let’s take the soft sets F_A and F_B in Example 3.2. Also, we take C = {e₂, e₃, e₄, e₆, e₇} and F_C = {(e₂, {u₁, u₂, u₄}), (e₃, {u₁, u₂, u₃}), (e₄, {u₃, u₄}), (e₆, {u₂, u₃}), (e₇, {u₁})}. Then we can say that F_A and F_B are ɛ-similar soft sets, wherever $ɛ \in (0, \frac{15}{49}]$ since $S (F_{A}, F_{B}) = \frac{15}{49}$ . Furthermore, F_A and F_C are significantly similar since $S (F_{A}, F_{C}) = \frac{5}{9}$ .

5 Symmetric difference of soft matrices

In this section, novel properties of the difference of soft matrices are depicted. Later on, the notion of symmetric difference of soft matrices are described.

Definition 5.1. ([11]) Let F_A, F_B ∈ S (U). Also, let [a_ij] and [b_ij] be soft matrices corresponding to the soft sets F_A and F_B, respectively. Then, the difference of [a_ij] and [b_ij] is a soft matrix [c_ij] such that c_ij = min {a_ij, 1 - b_ij} for all i, j. It is denoted by $[c_{ij}] = [a_{ij}] \tilde{\} [b_{ij}]$ .

Example 5.2. Let’s consider the soft sets F_A and F_B in Example 3.2. Then, the soft matrices corresponding the soft sets F_A and F_B are respectively $[a_{ij}] = [\begin{matrix} 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \end{matrix}]$ and $[b_{ij}] = [\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 \end{matrix}]$ . Then, the difference of the soft matrices [b_ij] and [a_ij] is the following soft matrix. $[c_{ij}] = [b_{ij}] \tilde{\} [a_{ij}] = [\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \end{matrix}] .$

Proposition 5.3. ([11]) Let F_A, F_B, F_C ∈ S (U). Also, let [a_ij], [b_ij] and [c_ij] be soft matrices corresponding to the soft sets F_A, F_B and F_C, respectively. [(i)] $[a_{ij}] \tilde{\} [a_{ij}] = [0]$ . [(ii)] $[a_{ij}] \tilde{\} [a_{ij}]^{c} = [a_{ij}]$ . [(iii)] $[a_{ij}] \tilde{\} [0] = [a_{ij}]$ and $[0] \tilde{\} [a_{ij}] = [0]$ . [(iv)] $[1] \tilde{\} [a_{ij}] = [a_{ij}]^{c}$ and $[a_{ij}] \tilde{\} [1] = [0]$ . [(v)] $[a_{ij}] \tilde{\} [b_{ij}] = [b_{ij}] \tilde{\} [a_{ij}] \Leftrightarrow [a_{ij}] = [b_{ij}]$ . [(vi)] $([a_{ij}] \tilde{\} [b_{ij}]) \tilde{\cap} [c_{ij}] = ([a_{ij}] \tilde{\cap} [c_{ij}]) \tilde{\} ([b_{ij}] \tilde{\cap} [c_{ij}])$ . [(vii)] $[a_{ij}] \tilde{\cap} ([b_{ij}] \tilde{\} [c_{ij}]) = ([a_{ij}] \tilde{\cap} [b_{ij}]) \tilde{\} ([a_{ij}] \tilde{\cap} [c_{ij}])$ . [(viii)] $([a_{ij}] \tilde{\} [b_{ij}]) \tilde{\} [c_{ij}] = ([a_{ij}] \tilde{\} [c_{ij}]) \tilde{\} ([b_{ij}] \tilde{\} [c_{ij}])$ .

Now, we describe novel properties for the difference of soft matrices.

Proposition 5.4. Let F_A, F_B, F_C ∈ S (U). Also, let [a_ij], [b_ij] and [c_ij] be soft matrices corresponding to the soft sets F_A, F_B and F_C, respectively. [(i)] $[a_{ij}] \tilde{\} [b_{ij}] = [a_{ij}] \tilde{\cap} [b_{ij}]^{c}$ . [(ii)] $([a_{ij}] \tilde{\cap} [b_{ij}]) \tilde{\} [c_{ij}] = ([a_{ij}] \tilde{\} [c_{ij}]) \tilde{\cap} ([b_{ij}] \tilde{\} [c_{ij}])$ . [(iii)] $([a_{ij}] \tilde{\cup} [b_{ij}]) \tilde{\} [c_{ij}] = ([a_{ij}] \tilde{\} [c_{ij}]) \tilde{\cup} ([b_{ij}] \tilde{\} [c_{ij}])$ . [(iv)] $[a_{ij}] \tilde{\} ([b_{ij}] \tilde{\cap} [c_{ij}]) = ([a_{ij}] \tilde{\} [b_{ij}]) \tilde{\cup} ([a_{ij}] \tilde{\} [c_{ij}])$ . [(v)] $[a_{ij}] \tilde{\} ([b_{ij}] \tilde{\cup} [c_{ij}]) = ([a_{ij}] \tilde{\} [{ab}_{ij}]) \tilde{\cap} ([a_{ij}] \tilde{\} [c_{ij}])$ .

Proof. Let [a_ij], [b_ij], [c_ij] ∈ SM (U).The proof of (i) is obvious, therefore omitted.(ii) Assume that $[a_{ij}] \tilde{\cap} [b_{ij}] = [d_{ij}]$ where d_ij = min {a_ij, b_ij} for all i, j. And suppose that $[d_{ij}] \tilde{\} [c_{ij}] = [g_{ij}]$ where g_ij = min {d_ij, 1 - c_ij} = min {min {a_ij, b_ij}, 1 - c_ij} for all i, j.We now illustrate the right-hand side of equality. Assume that $[a_{ij}] \tilde{\} [c_{ij}] = [h_{ij}]$ where h_ij = min {a_ij, 1 - c_ij} for all i, j, and also $[b_{ij}] \tilde{\} [c_{ij}] = [k_{ij}]$ where k_ij = min {b_ij, 1 - c_ij} for all i, j. Suppose that $[h_{ij}] \tilde{\cap} [k_{ij}] = [l_{ij}]$ where l_ij = min {h_ij, k_ij} = min {min {a_ij, 1 - c_ij}, min {b_ij, 1 - c_ij}} = min {min {a_ij, b_ij}, 1 - c_ij} for all i, j. Since g_ij = l_ij for all i, j, the proof is completed.(iii)-(v) The proofs can be proved similar to (ii).□

Now, we describe the symmetric difference of the soft matrices as follows.

Definition 5.5. Let F_A, F_B ∈ S (U). Also, let [a_ij] and [b_ij] be soft matrices corresponding to the soft sets F_A and F_B, respectively. Then, the symmetric difference of [a_ij] and [b_ij] is a soft matrix [c_ij] such that c_ij = min {max {a_ij, b_ij}, 1 - min {a_ij, b_ij}} for all i, j and it is denoted by $[c_{ij}] = [a_{ij}] \tilde{△}$ .

Example 5.6. Let’s consider again the soft matrices [a_ij] and [b_ij] in Example 5.2. Then, the symmetric difference of these soft matrices is the following soft matrix. $[c_{ij}] = [a_{ij}] \tilde{△} [b_{ij}] = [\begin{matrix} 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \end{matrix}] .$

Proposition 5.7. Let F_A, F_B, F_C ∈ S (U). Also, let [a_ij], [b_ij] and [c_ij] be soft matrices corresponding to the soft sets F_A, F_B and F_C respectively. [(i)] $[a_{ij}] \tilde{△} [a_{ij}] = [0]$ . [(ii)] $[a_{ij}] \tilde{△} [a_{ij}]^{c} = [1]$ . [(iii)] $[a_{ij}] \tilde{△} [0] = [a_{ij}]$ . [(iv)] $[1] \tilde{△} [a_{ij}] = [a_{ij}]^{c}$ . [(v)] $[a_{ij}] \tilde{△} [b_{ij}] = [b_{ij}] \tilde{△} [a_{ij}]$ . [(vi)] $[a_{ij}] \tilde{△} [b_{ij}] = ([a_{ij}] \tilde{\} [b_{ij}]) \tilde{\cup} ([b_{ij}] \tilde{\} [a_{ij}])$ . [(vii)] $[a_{ij}] \tilde{△} [b_{ij}] = [a_{ij}]^{c} \tilde{△} [b_{ij}]^{c}$ . [(viii)] $([a_{ij}] \tilde{△} [b_{ij}]) \tilde{△} [c_{ij}] = [a_{ij}] \tilde{△} ([b_{ij}] \tilde{△} [c_{ij}])$ .

Proof. Let [a_ij], [b_ij], [c_ij] ∈ SM (U).The proofs of (i)-(v) are intuitively clear, therefore omitted.(vi) Assume that $[a_{ij}] \tilde{△} [b_{ij}] = [d_{ij}]$ where d_ij = min {max {a_ij, b_ij}, 1 - min {a_ij, b_ij}} for all i, j.Now consider the right-hand side of equality. Assume that $[a_{ij}] \tilde{\} [b_{ij}] = [g_{ij}]$ where g_ij = min {a_ij, 1 - b_ij} for all i, j, and also $[b_{ij}] \tilde{\} [a_{ij}] = [h_{ij}]$ where h_ij = min {b_ij, 1 - a_ij} for all i, j. Suppose that $[g_{ij}] \tilde{\cup} [h_{ij}] = [k_{ij}]$ where k_ij = max {g_ij, h_ij} = max {min {a_ij, 1 - b_ij}, min {b_ij, 1 - a_ij}} = min {max {a_ij, b_ij}, 1 - min {a_ij, b_ij}} for all i, j. Since d_ij = k_ij for all i, j, the proof is completed.(vii)-(viii) The proofs can be proved similar to (vi).□

6 Similarity measure between two soft matrices

In this section, the similarity measure between two soft matrices is introduced, and then its Scilab code is presented.

Definition 6.1. Let F_A, F_B ∈ S (U). Also, let [a_ij] and [b_ij] be soft matrices corresponding to the soft sets F_A and F_B, respectively. Then, the similarity measure between these soft matrices is defined and denoted by

$S_{m} ([a_{ij}], [b_{ij}]) = \frac{| A \cap B |}{| A \cup B |} (1 - \frac{1}{n | A \cup B |} \sum_{j = 1}^{m} \sum_{i = 1}^{n} c_{ij})$ ,

where $[c_{ij}] = [a_{ij}] \tilde{△} [b_{ij}]$ .

Now, we give Scilab code, which speeds up the process of calculations, for the similarity measure of soft matrices (see Table 1).

Table 1

x=input(‘enter the value of |A ∩ B|’)

y=input(‘enter the value of |A ∪ B|’)

a=input(‘enter soft matrix1’)

b=input(‘enter soft matrix2’)

function Sm=sim(a,b)

[n,m]=size(a);

[n,m]=size(b);

c=zeros(n,m);

for i=1:n

for j=1:m

c(i,j)=min(max(a(i,j),b(i,j)),1-min(a(i,j),b(i,j)));

end

end

Sm=(x/y)(1-((1/(ny))*sum(sum(c))));

endfunction

x=input(‘enter the value of \|A ∩ B\|’)
function Sm=sim(a,b)
[n,m]=size(a);
[n,m]=size(b);
c=zeros(n,m);
for i=1:n
for j=1:m
c(i,j)=min(max(a(i,j),b(i,j)),1-min(a(i,j),b(i,j)));
end
end
Sm=(x/y)(1-((1/(ny))*sum(sum(c))));
endfunction

Consider the soft sets F_A and F_B for each part in Example 4.2. When the soft matrices corresponding to these soft sets are created, the same results as in Example 4.2 can be obtained by using the Definition 6.1.

Note: The results of similarity measures of soft sets and corresponding soft matrices are the same.

Also, we present a new example for the similarity measure of soft matrices.

Example 6.2. Let U = {u₁, u₂, u₃, u₄, u₅} be a universal set, E = {e₁, e₂,…, e₂₀} be a set of parameters and A = E - {e₁₀, e₁₂, e₁₈}, B = E - {e₄, e₅, e₆, e₁₂, e₁₅, e₁₆, e₁₈}. Also, let

F_A = {(e₁, {u₁, u₂, u₃}), (e₂, {u₂, u₃}), (e₃, {u₁, u₃}), (e₄, {u₁}), (e₅, {u₅}), (e₆, {u₂, u₃, u₄}), (e₇, {u₁}), (e₉, {u₂, u₃}), (e₁₁, U), (e₁₃, {u₂, u₄}), (e₁₅, U), (e₁₆, {u₂, u₃, u₄}), (e₁₇, U), (e₁₉, {u₁}), (e₂₀, {u₂, u₄})},

F_B = {(e₁, {u₁, u₄}), (e₂, {u₂, u₃}), (e₃, {u₂, u₄}), (e₇, {u₂, u₃}), (e₈, {u₃}), (e₉, {u₁, u₃}), (e₁₀, {u₅}), (e₁₁, U), (e₁₃, {u₂, u₃, u₄}), (e₁₉, {u₁}), (e₂₀, {u₁, u₂})}.

Then, the soft matrices of these soft sets are respectively

$[a_{ij}] = [\begin{matrix} 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \end{matrix}]$ and $[b_{ij}] = [\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]$ . Thus, we obtain the symmetric difference of these soft matrices as follows $[c_{ij}] = [a_{ij}] \tilde{△} [b_{ij}] = [\begin{matrix} 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \end{matrix}]$ . From Definition 6.1, we calculate that $S_{m} ([a_{ij}], [b_{ij}]) = \frac{2}{3} (1 - \frac{7}{18}) = \frac{11}{27}$ .

As in the above example, the implementation of symmetric difference operation for the soft sets containing many parameters or alternatives is more difficult than that of the soft matrices. Therefore, the soft matrices provide enormous convenience for such situations.

Proposition 6.3. Let F_A, F_B ∈ S (U). Also, let [a_ij] and [b_ij] be soft matrices corresponding to the soft sets F_A and F_B, respectively. Then, we have [(i)] S_m ([a_ij], [a_ij]) =1. [(ii)] S_m ([a_ij], [a_ij] ^c) =0. [(iii)] S_m ([a_ij], [b_ij]) = S_m ([b_ij], [a_ij]). [(iv)] 0 ≤ S_m ([a_ij], [b_ij]) ≤1.

Proof. The results are obvious from the proof of Proposition 4.3.□

Proposition 6.4. Let $F_{A} \tilde{\subseteq} F_{B} \tilde{\subseteq} F_{C}$ such that A ⊆ B ⊆ C ⊆ E. Also, let [a_ij], [b_ij] and [c_ij] be soft matrices corresponding to the soft sets F_A, F_B and F_C, respectively. Then, we have S_m ([a_ij], [c_ij]) ≤ S_m ([b_ij], [c_ij]).

Proof. The result is obvious from the proof of Proposition 4.4.□

Definition 6.5 Let [a_ij], [b_ij] ∈ SM (U). Then, [a_ij] and [b_ij] are said to be ɛ-similar soft matrices if and only if S_m ([a_ij], [b_ij]) ≥ ɛ for ɛ ∈ (0, 1). These two soft matrices are said to be significantly similar if $S_{m} ([a_{ij}], [b_{ij}]) > \frac{1}{2}$ .

7 Applications

In this section, the outstanding examples are given to demonstrate the practicality and effectivity of similarity measure between the soft matrices.

Example 7.1. Suppose that an automotive company wants to hire an automotive engineer. There are three candidates who apply to the company. In line with this objective, according to the parameters determined by the company, five different automobiles of the company must be evaluated by three candidates and an engineer who already works in the company. As a result of this evaluation, the company plans to hire the candidate who has the closest opinion to the company’s engineer. Assume that the set of the evaluated automobiles is U = {u₁, u₂, u₃, u₄, u₅} and the set of evaluation parameters is E = {e₁, e₂,…, e₁₀}, where e_j stand for “comfort”, “design-aesthetic”, “fuel consumption”, “image-prestige”, “performance”, “engine power”, “equipment”, “cylinder volume”, “ safety” and “road handling”. After evaluating these automobiles, the company has the following evaluation results.Candidate 1: For A₁ = E, F_{A
₁} = {(e₁, {u₂, u₃}), (e₂, {u₅}), (e₃, {u₁, u₃, u₅}), (e₄, {u₁, u₂, u₄, u₅}), (e₅, {u₁, u₂, u₄, u₅}), (e₆, {u₂, u₃, u₅}), (e₇, U), (e₈, {u₁, u₄}), (e₉, {u₁, u₃, u₄}), (e₁₀, {u₁, u₃, u₅})}. Candidate 2: For A₂ = E, F_{A
₂} = {(e₁, {u₂, u₃}), (e₂, {u₂}), (e₃, {u₂, u₃}), (e₄, {u₂, u₃, u₄}), (e₅, {u₁, u₃, u₅}), (e₆, {u₂, u₅}), (e₇, {u₁, u₂, u₃, u₅}), (e₈, U), (e₉, {u₁, u₃, u₄, u₅}), (e₁₀, {u₂, u₃, u₅})}. Candidate 3: For A₃ = E, F_{A
₃} = {(e₁, {u₁, u₂, u₄}), (e₂, {u₂, u₅}), (e₃, {u₁, u₃, u₄}), (e₄, {u₁, u₃}), (e₅, {u₁, u₂, u₅}), (e₇, {u₁, u₂u₄}), (e₈, U), (e₉, {u₂, u₃}), (e₁₀, {u₁, u₂, u₃, u₅})}. The company’s engineer: For B = E, F_B = {(e₁, {u₃, u₄, u₅}), (e₂, {u₁, u₃, u₄}), (e₃, {u₁, u₂, u₄}), (e₄, U), (e₅, {u₂, u₃, u₄}), (e₆, {u₂, u₃}), (e₇, U), (e₈, {u₁, u₂, u₄, u₅}), (e₉, U), (e₁₀, {u₁, u₃, u₅})}. To facilitate calculations, we present the soft matrices corresponding to these soft sets, respectively. $[a_{ij}^{1}] = [\begin{matrix} 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 1 \end{matrix}]$ , $[a_{ij}^{2}] = [\begin{matrix} 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}]$ , $[a_{ij}^{3}] = [\begin{matrix} 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 \\ 1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 \end{matrix}]$ and $[b_{ij}] = [\begin{matrix} 0 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \end{matrix}]$ . During calculating the similarity measure $S_{m} ([a_{ij}^{k}], [b_{ij}])$ between each $[a_{ij}^{k}]$ and [b_ij], the company has $S_{m} ([a_{ij}^{1}], [b_{ij}]) = \frac{3}{5}$ , $S_{m} ([a_{ij}^{2}], [b_{ij}]) = \frac{27}{50}$ and $S_{m} ([a_{ij}^{3}], [b_{ij}]) = \frac{23}{50}$ . Then, the similarity measure $S_{m} ([a_{ij}^{1}], [b_{ij}])$ between $[a_{ij}^{1}]$ and [b_ij] is the largest one. Therefore, Candidate 1 is the closest to the opinion of company’s engineer. So, it is the right decision to hire him/her.

Table 2

Paper Problem Similarity measures Results

Majumdar and Samanta’s Definition 3.1 in [15] 1

Kharal’s Definition 23 in [12] 2^*

[22] For the soft sets F and G in the Example 1 Kharal’s Definition $(S_{K}^{e})$ in [12] 1

Yang’s Definition 6 in [22] (1, 1)

Our Definition 1

Majumdar and Samanta’s Definition 3.1 in [15] ${\frac{0}{0}}^{}$

Kharal’s Definition 23 in [12] 1

[12] For the soft set F₁ in the Example 8 Kharal’s Definition $(S_{K}^{e})$ in [12] 1

Yang’s Definition 6 in [22] $(1, \frac{0}{0})^{}$

Our Definition 1

Majumdar and Samanta’s Definition 3.1 in [15] $\frac{1}{8}$

Kharal’s Definition 23 in [12] ${\frac{9}{8}}^{}$

[15] For the soft sets F and G in the Application Kharal’s Definition $(S_{K}^{e})$ in [12] $\frac{1}{1 + \sqrt{14}}$

Yang’s Definition 6 in [22] $(1, \frac{1}{8})$

Our Definition $\frac{1}{8}$

Majumdar and Samanta’s Definition 3.1 in [15] $\frac{3}{4}$

Kharal’s Definition 23 in [12] ${\frac{7}{4}}^{}$

[15] For the soft sets F and H in the Application Kharal’s Definition $(S_{K}^{e})$ in [12] ${\frac{1}{3}}^{†}$

Yang’s Definition 6 in [22] $(1, \frac{3}{4})$

Our Definition $\frac{3}{4}$

Paper	Problem	Similarity measures	Results
		Majumdar and Samanta’s Definition 3.1 in [15]	1
		Kharal’s Definition 23 in [12]	2^*
[22]	For the soft sets F and G in the Example 1	Kharal’s Definition $(S_{K}^{e})$ in [12]	1
		Yang’s Definition 6 in [22]	(1, 1)
		Our Definition	1
		Majumdar and Samanta’s Definition 3.1 in [15]	${\frac{0}{0}}^{*}$
		Kharal’s Definition 23 in [12]	1
[12]	For the soft set F₁ in the Example 8	Kharal’s Definition $(S_{K}^{e})$ in [12]	1
		Yang’s Definition 6 in [22]	$(1, \frac{0}{0})^{*}$
		Our Definition	1
		Majumdar and Samanta’s Definition 3.1 in [15]	$\frac{1}{8}$
		Kharal’s Definition 23 in [12]	${\frac{9}{8}}^{*}$
[15]	For the soft sets F and G in the Application	Kharal’s Definition $(S_{K}^{e})$ in [12]	$\frac{1}{1 + \sqrt{14}}$
		Yang’s Definition 6 in [22]	$(1, \frac{1}{8})$
		Our Definition	$\frac{1}{8}$
		Majumdar and Samanta’s Definition 3.1 in [15]	$\frac{3}{4}$
		Kharal’s Definition 23 in [12]	${\frac{7}{4}}^{*}$
[15]	For the soft sets F and H in the Application	Kharal’s Definition $(S_{K}^{e})$ in [12]	${\frac{1}{3}}^{†}$
		Yang’s Definition 6 in [22]	$(1, \frac{3}{4})$
		Our Definition	$\frac{3}{4}$

In the table, * means that the result is undefined. † means that the result is counterintuitive.

Example 7.2. The similarity measure of soft matrices can be used to detect whether a patient is suffering from a certain disease or not. An enterprise is made to forecast the possibility that a patient having certain visible symptoms is suffering from prostate cancer. In this direction, a model soft set for prostate cancer and the soft sets for the patients are constructed. Immediately after the soft matrices corresponding to the soft sets are constructed and the similarity measure is calculated. It is diagnosed that the patient is possibly suffering from prostate cancer if the similarity measure of soft matrices is $ɛ \geq \frac{3}{5}$ .Assume that the universal set consists of only two elements which are yes and no, that is, U = {y, n}. E = {e₁, e₂, e₃, e₄, e₅, e₆, e₇} is the set of parameters which are the symptoms. These symptoms are “trouble urinating”, “throwing up”, “decrease in the stream of urine”, “blood in semen”, “stomachache” “discomfort in the pelvic area”, “erectile dysfunction” and “pyrexia”, respectively.When talking to two patients having the diseases and pains, the following soft sets are constructed.First patient’s soft set: F_{A
₁} = {(e₁, {y}), (e₂, {y}), (e₃, {y}), (e₄, {y}), (e₅, {y}), (e₆, {n}), (e₇, {y}), (e₈, {y})}. Second patient’s soft set: F_{A
₂} = {(e₁, {y}), (e₂, {n}), (e₃, {y}), (e₄, {y}), (e₅, {y}), (e₆, {y}), (e₇, {y}), (e₈, {y})}. Also, a model soft set created with the help of a urologist for prostate cancer: F_B = {(e₁, {y}), (e₂, {n}), (e₃, {y}), (e₄, {y}), (e₅, {n}), (e₆, {y}), (e₇, {y}), (e₈, {n})}. Now, we respectively present the soft matrices corresponding to these soft sets as follows: $[a_{ij}^{1}] = [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \end{matrix}]$ , $[a_{ij}^{2}] = [\begin{matrix} 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}]$ and $[b_{ij}] = [\begin{matrix} 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \end{matrix}]$ After the similarity measure $S_{m} ([a_{ij}^{k}], [b_{ij}])$ between each $[a_{ij}^{k}]$ and [b_ij] are calculated, it is obtained that $S_{m} ([a_{ij}^{1}], [b_{ij}]) = \frac{1}{2}$ and $S_{m} ([a_{ij}^{2}], [b_{ij}]) = \frac{3}{4}$ .

Hence, it is said that the second patient is possibly suffering from the prostate cancer since the similarity measure $S_{m} ([a_{ij}^{2}], [b_{ij}]) = \frac{3}{4} > \frac{3}{5}$ between $[a_{ij}^{2}]$ and [b_ij]. However, it is said that the first patient is not possibly suffering from the prostate cancer.

Comparison: In this part, we compare the results of our similarity measure with the results of some of the existing similarity measures in the literature. We point out that our similarity measure gives more satisfactory results than the others. The results of these comparisons are displayed in the Table 2.

As it can be seen from the Table 2, each of the existing similarity measures have various problems in getting the similarity of some soft sets. However, our similarity measure gives the consistent results for these soft sets.

8 Conclusion

In this study, we discussed new operations of the soft sets and the soft matrices. By utilizing these notions, we presented novel similarity measures for the soft sets and the soft matrices. We pointed out that the results of similarity measures of soft sets and soft matrices are the same. Also, the Scilab code was provided so that the similarity measure can be easily transferred to the computer via matrices. Thus, we overcame some difficulties and constraints which are encountered in finding the similarity measure of soft sets.

It is worth mentioning that the ones presented here will shed light on future work, especially on the soft symmetric difference. Also, the computation of similarity measure of soft sets with the help of matrices will present a new perspective to next steps.

References

Ali

M.I.

, Feng

, Liu

, Min

W.K.

, Shabir

, On some new operations in soft set theory, Comput Math Appl57 (2009), 1547–1553.

Alshehri

N.O.

, Akram

, Al-ghami

R.S.

, Applications of soft sets in K-algebras, Adv Fuzzy Syst2013 (2013), 8.

Atagün

A.O.

, Kamacı

, Oktay

, Reduced soft matrices and generalized products with applications in decision making, Neural Comput Appl29 (2018), 445–456.

Babitha

K.V.

, Sunil

J.J.

, Soft set relations and functions, Comput Math Appl60 (2010), 1840–1849.

Basu

T.M.

, Mahapatra

N.M.

, Mondal

S.K.

, Matrices in soft set theory and their applications in decision making problems, South Asian J Math2 (2012), 126–143.

Borah

M.J.

, Hazarika

, Some applications of similarity of soft sets, J Intell Fuzzy Syst33 (2017), 3763–3777.

Cagman

and Enginoglu

, Soft matrix theory and its decision making, Comput Math Appl59 (2010), 3308–3314.

Cagman

and Enginoglu

, Soft set theory and uni-int decision making, Eur J Oper Res207 (2010), 848–855.

Feng

, Jun

Y.B.

, Zhao

, Soft semirings, Comput Math Appl56 (2008), 2621–2628.

10.

Kamacı

, Atagün

A.O.

, Sönmezoglu

, Row-products of soft matrices with applications in multiple-disjoint decision making, Appl Soft Comput62 (2018), 892–914.

11.

Kamacı

, Atagün

A.O.

, Aygün

, Difference operations of soft matrices with applications in decision making, Punjab Univ J Math (accepted).

12.

Kharal

, Distance and similarity measures for soft sets, New Math Nat Comput6 (2010), 321–334.

13.

Maji

P.K.

, Biswas

, Roy

A.R.

, Soft set theory, Comput Math Appl45 (2003), 555–562.

14.

Maji

P.K.

, Roy

A.R.

, Biswas

, An application of soft sets in a decision making problem, Comput Math Appl44 (2002), 1077–1083.

15.

Majumdar

, Samanta

S.K.

, Similarity measure of soft sets, New Math Nat Comput4 (2008), 1–12.

16.

Molodtsov

, Soft set theory-first results, Comput Math Appl37 (1999), 19–31.

17.

Pei

, Miao

From soft sets to information systems, Proceedings of Granular Computing, IEEE, 2 (2005), 617–621 Hu

, Liu

, Skowron

, Lin

T.Y.

, Yager

R.R.

and Zhang

18.

Sezgin

, Atagün

A.O.

, On operations of soft sets, Comput Math Appl61 (2011), 1457–1467.

19.

Sezgin

, Atagün

A.O.

, Aygün

, A note on soft nearrings and idealistic soft near-rings, Filomat25 (2011), 53–68.

20.

Yang

H.-L.

, Guo

Z.-L.

, Kernels and clousures of soft set relations, and soft set relation mappings, Comput Math Appl61 (2011), 651–662.

21.

Yang

H.-L.

, Liao

, Li

S.-G.

, On soft continuous mappings and soft connectedness of soft topological spaces, Hacet J Math Stat44 (2015), 385–398.

22.

Yang

, New similarity measures for soft sets and their application, Fuzzy Inf Eng1 (2013), 19–25.

23.

Zhu

, Wen

, Operations of soft set revisited, J Appl Math2013 (2013), 7.