Type-2 soft sets

1 Introduction

Uncertainty is the most prevalent aspect in the natural occurrence of events. Therefore the need to device out tools in order to deal with such phenomena became a necessity. The introduction of the theory of Fuzzy Sets by Zadeh [28] in 1965 brought about a paradigmatic change in the approach of dealing with uncertainties, which until then was dealt, to some extent, by the probability theory, in the form of randomness. However, the theory of fuzzy sets which was designed to model uncertainties, had certain shortcomings since the membership function of a fuzzy set is certain once the parameters are specified [19]. Quite naturally several other mathematical structures came into existence in the form of improvisations of the existing fuzzy set theory viz. the theory of L-Fuzzy sets by Goguen in 1967 [10], the interval-valued fuzzy sets by Zadeh in 1975 [29], the concept of type 2 fuzzy Sets by L.A. Zadeh in 1975 [29], the theory of rough sets by Pawlak in 1982 [24], the theory of intuitionistic fuzzy sets by K.T. Atanassov [2], to name a few. Interestingly, as pointed out by Zadeh [29], the notion of type 2 fuzzy sets traces its way back to the close association between the “concept of linguistic truth on one hand and fuzzy sets on the other hand where the grades of membership are specified in linguistic terms” thereby considering fuzziness of higher order. Type 2 fuzzy sets are characterized by fuzzy membership functions that are themselves type 1 fuzzy in nature. However, the difficulties encountered in comprehending the three dimensional nature of type 2 fuzzy sets and in deriving formulae for union, intersection etc. using Zadeh’s Extension Principle constitute the major complexities encountered while dealing with the theory of type 2 fuzzy sets.

All the afore-mentioned theories had their own limitations and fell short in catering fully to the needs of handling uncertainty, which according to Molodtsov [21] was possibly due to the lack of a “parameterization tool” who came up with the theory of “Soft Sets” in dealing with intrinsic imprecision using adequate parameterization. In his work, Molodtsov pointed out that Zadeh’s fuzzy sets were special types of soft sets, the parameter set being the unit interval [0, 1]. In this respect it might be stated that the parameterization technique of soft set theory is more user friendly as compared to the membership function approach of fuzzy set theory in the field of their applications in real life problems. Ever since then the theory of soft sets has undergone rapid developments. Maji et al. [15] presented a detailed theoretical study on soft sets. Thereafter, the theory of soft sets has been enriched by the contribution of works of authors such as the introduction of soft groups by Aktas and Cagman [1] and the study of soft topological spaces by Shabir and Naz [26]. Soft metric, soft metric spaces and soft inner product spaces were studied by Das and Samanta [5–7]. Soft mappings, similarity of soft sets and soft set entropy were defined by Majumdar and Samanta [16–18] whereas Kharal et al. [12] defined mappings on soft classes. At present, works on applications of soft set theory are progressing rapidly. Maji and Roy [14] proposed a parameterized reduction of soft sets and applied it to a decision making problem. Chen et al. [4] introduced a new definition for the parameterized reduction of soft sets. Pei and Miao [25] showed that soft sets are a special type of information system. Cagman and Enginoglu [3] defined uni-int operators and their application in decision making. Other applications of soft sets include the study of texture classification [22] and data analysis [30] using softtheory.

Observing such a huge potential of the Soft Set Theory, it is natural to investigate the possibility of existence of a parameterized structure for type 2 fuzzy sets as well. Since the initial approach to cater to this need with the help of ordinary soft sets fell short, our work at hand aims to initiate the concept of type-2 soft sets which is a generalization of the concept of Molodtsov’s soft sets in the sense that it involves parameterization over an already parameterized set and hence has more freedom and efficiency compared to usual soft sets (which may be termed as type-1 soft sets) in handling impreciseness.

The organization of the paper is as follows:

Some useful preliminary results have been stated in Section 2; Section 3 introduces the notion of type-2 soft sets where it has been shown that type 2 fuzzy sets are special types of type-2 soft sets; in Section 4, a detailed theoretical study has been carried out in the form of defining various operations over type-2 soft sets; in Section 5, the definitions of the image and inverse image of a type-2 soft set under a mapping have been stated and some properties related to those definitions have been studied; Section 6 deals with an application of type-2 soft sets in a decision making problem while Section 7 concludes the paper.

2 Preliminaries

Definition 2.1. [15] Let U be an initial universe and E be a set of parameters. Let P ( U ) denotes the power set of U and A ⊂ E. A pair (F, A) is called a soft set if and only if F is a mapping of A into P ( U ).

Example 2.2. [15] Suppose U is the set of houses under consideration, E is the set of parameters, each parameter being a word, a phrase or a sentence. Suppose that there are six houses in the universe, given by U = { h 1 , h 2 , h 3 , h 4 , h 5 , h 6 } and E = { expensive ; beautiful ; wooden ; cheap ; in green surroundings ; modern ; in good repair ; in bad repair } be the set of parameters. Let, A = { expensive ; beautiful ; wooden ; cheap ; in green surroundings }

Then, A ⊂ E.

Suppose that, F : A → P ( U ) is given by, F ( expensive ) = { h 2 , h 4 } F ( beautiful ) = { h 1 , h 3 } F ( wooden ) = { h 3 , h 4 , h 5 } F ( cheap ) = { h 1 , h 3 , h 5 } F ( in green surroundings ) = { h 1 }

Then (F, A) is a soft set under consideration and it is represented as, ( F , A ) = { ( expensive , { h 2 , h 4 } ) , ( beautiful , { h 1 , h 3 } ) , ( wooden , { h 3 , h 4 , h 5 } ) ( cheap , { h 1 , h 3 , h 5 } ) , ( in green surroundings , { h 1 } ) } .

Definition 2.3. [8] For two soft sets (F, A) and (G, B) over a common universe U, we say that (F, A) is a soft subset of (G, B) if,

A ⊂ B, and

This is represented as, ( F , A ) ⊂ ˜ ( G , B ).

Definition 2.4. [15] Two soft sets (F, A) and (G, B) over a common universe are said to be soft equal if (F, A) is a soft subset of (G, B) and (G, B) is a soft subset of (F, A).

Definition 2.5. [16] The complement of a soft set (F, A) is denoted by (F, A)  ^c and is defined by (F, A)  ^c  = (F ^c , A), where F c : A → P ( U ) is a mapping given by F c ( e ) = { F ( e ) } c = U - F ( e ) , ∀ e ∈ A .

Definition 2.6. [15] A soft set (F, A) over U is said to be a null soft set, denoted by Φ, if ∀ e ∈ A , F ( e ) = φ the null set .

Definition 2.7. [15] A soft set (F, A) over U is said to be an absolute soft set, denoted by A ˜, if ∀ e ∈ A , F ( e ) = U , the universal set .

Proposition 2.8. [15]

    A ˜ c = Φ

Definition 2.9. [15] If (F, A) and (G, B) are two soft sets then “(F, A)  AND  (G, B)”, denoted by (F, A) ∧ (G, B), is defined by, ( F , A ) ∧ ( G , B ) = ( H , A × B ) , where H ( α , β ) = F ( α ) ∩ G ( β ) , ∀ ( α , β ) ∈ A × B .

Definition 2.10. [15] If (F, A) and (G, B) are two soft sets then “(F, A)  OR  (G, B)”, denoted by (F, A) ∨ (G, B), is defined by, ( F , A ) ∨ ( G , B ) = ( 0 , A × B ) , where 0 ( α , β ) = F ( α ) ∪ G ( β ) , ∀ ( α , β ) ∈ A × B .

Remark 2.11. The “AND” and “OR” operations over soft sets are associative as well as distributive.

Proposition 2.12. [15]

((F, A) ∨ (G, B))  ^c  = (F, A)  ^c  ∧ (G, B)  ^c

Definition 2.13. [15] The union of two soft sets (F, A) and (G, B) over the common universe U is a soft set (H, C), denoted by, ( F , A ) ∪ ˜ ( G , B ) = ( H , C ), where C = A ∪ B, and ∀e ∈ C, H ( e ) = { F ( e ) , if e ∈ A - B G ( e ) , if e ∈ B - A F ( e ) ∪ G ( e ) , if e ∈ A ∩ B

Definition 2.14. [8] The intersection of two soft sets (F, A) and (G, B) over the common universe U is a soft set (H, C), denoted by, ( F , A ) ∩ ˜ ( G , B ) = ( H , C ) , where C = A ∩ B , and ∀ e ∈ C , H ( e ) = F ( e ) ∩ G ( e ) .

Remark 2.15. For any two soft sets (F, A) and (G, B), defined over the soft universe (X,  E), De-Morgan’s Laws do not hold in general, however, it follows that, ( ( F , A ) ∪ ˜ ( G , B ) ) c ⊃ ˜ ( F , A ) c ∩ ˜ ( G , B ) c ( ( F , A ) ∩ ˜ ( G , B ) ) c ⊂ ˜ ( F , A ) c ∪ ˜ ( G , B ) c

But if the parameter sets are taken to be the same then De-Morgan’s Laws hold. viz. ( ( F , A ) ∪ ˜ ( G , A ) ) c = ( F , A ) c ∩ ˜ ( G , A ) c ( ( F , A ) ∩ ˜ ( G , A ) ) c = ( F , A ) c ∪ ˜ ( G , A ) c

Proposition 2.16. [15]

( F , A ) ∪ ˜ ( F , A ) = ( F , A )

Definition 2.17. [23] Let X and Y be two non-empty sets and f : X → Y be a mapping. Then,

the image of a soft set (F, A) ∈ S (X,  A) ,  S (X,  A) being the collection of all soft sets over the soft universe (X,  A), under the mapping f is defined by f ( ( F , A ) ) = ( f ( F ) , A ) , where [f (F)] (α) = f [F (α)] , ∀ α ∈ A.

Proposition 2.18. [23] Let X and Y be two non-empty sets and f : X → Y be a mapping. If (F ₁,  A) , (F ₂,  A) ∈ S (X,  A) then

( F 1 , A ) ⊆ ˜ ( F 2 , A ) ⇒ f ( F 1 , A ) ⊆ ˜ f ( F 2 , A )

Proposition 2.19. [23] Let X and Y be two non-empty sets and f  :  X  →  Y be a mapping. If (G ₁,  A) , (G ₂,  A) ∈ S (Y,  A) then

( G 1 , A ) ⊆ ˜ ( G 2 , A ) ⇒ f - 1 ( G 1 , A ) ⊆ ˜ f - 1 ( G 2 , A )

Proposition 2.20. [23] Let X and Y be two non-empty sets and f : X → Y be a mapping.

If (F, A) ∈ S (X, A) then ( F , A ) ⊆ ˜ f - 1 [ f [ ( F , A ) ] ]and equality occurs if f is injective.

Definition 2.21. [29] A fuzzy set is of type n,  n = 2, 3, …, if its membership function ranges over fuzzy sets of type n - 1. The membership function of a fuzzy set of type 1 ranges over the interval [0,1].

3 Type-2 soft sets

In this section we are introducing a concept of type-2 soft sets (T2SS, in short) and from now on we refer to Molodtsov’s soft sets as type-1 soft sets (T1SS, in short).

Definition 3.1. Let (X, E) be a soft universe and S (X) be the collection of all type-1 soft sets over (X, E). Then a mapping F : A → S (X) ,  A ⊂ E is called a type-2 soft set over (X, E) and it is denoted by [ F , A ].

In this case, corresponding to each parameter e ∈ A , F ( e ) is a type-1 soft set. Thus, for each e ∈ A, there exists a T1SS, (F _e , S _e ) such that F ( e ) = ( F e , S e ) where F _e  : S _e  → p (X) and S _e  ⊂ E. In this case, we refer to the parameter set A as the “primary set of parameters”, while the set of parameters ∪S _e is known as the “underlying set of parameters”.

Example 3.2. Consider an example of a diet chart. Let the universe X under consideration be the set of some food items where X = { rice , pudding , noodles , ice cream , chicken soup , chicken sandwich }

Let E be the set of parameters defined as, E = { solid diet , soft diet , fluid diet , calorie rich diet , fibre rich diet , protein rich diet , carbohydrate rich diet } Suppose, A = { calorie rich diet , fibre rich diet , carbohydrate rich diet }

Then, A ⊂ E.

Let [ F , A ] be a type-2 soft set over X, which denotes the characteristics of the above mentioned food items in terms of food value. Suppose, F : A → S (X) be defined as F ( calorie rich diet ) = { solid diet { rice , noodles } , soft diet { pudding , ice cream } } F ( fibre rich diet ) = { solid diet { chicken sandwich , noodles } , soft diet { chicken soup } } F ( carbohydrate rich diet ) = { solid diet { rice } }

The above mentioned type 2 soft set is interpreted as follows:

Out of the various food items under speculation, “calorie rich, solid diet” are rice and noodles, “calorie rich, soft diet” include pudding and ice cream, “fibre rich, solid diet” constitute chicken sandwich and noodles, “fibre rich, soft diet” is chicken soup and “carbohydrate rich, solid diet” is rice.

Here, the set of parameters A constitutes the primary set of parameters whereas the set {solid diet,  soft diet} form the set of underlying parameters.

Remarks 3.3.

Type 2 fuzzy sets introduced by L.A. Zadeh may be considered as special types of type-2 soft sets when the parameters are considered over [0, 1].

Let X be the universe under consideration.

The point-valued representation of a type-2 fuzzy set [20], denoted by A ¯, is given by, A ¯ = { ( x , u ) , μ A ¯ ( x , u ) | ∀ x ∈ X , ∀ u ∈ J x ⊆ [ 0 , 1 ] } where μ A ¯ ( x , u ) is the type 2 membership function for all x ∈ X.

Then, for any α ∈ [0, 1], the corresponding α-plane of A ¯ is given by, A ¯ α = { ( x , u ) , μ A ¯ ( x , u ) ≥ α | ∀ x ∈ X , ∀ u ∈ J x ⊆ [ 0 , 1 ] } For a particular α ∈ [0, 1], construct the set, S α = { u : ( x , u ) ∈ A ¯ α }

Also define a mapping F α : S α → P ( X ) by, F α ( u ) = { x : ( x , u ) ∈ A ¯ α } , u ∈ S α .

Thus, (F _α ,  S _α ) constitutes a type-1 soft set over (X,   [0, 1]).

Next, define a mapping F : [ 0 , 1 ] → S ( X , [ 0 , 1 ] )by, F ( α ) = ( F α , S α )

Thus, ( F , [ 0 , 1 ] ) is a type-2 soft set over (X,   [0, 1]).

Example 3.4. Considering the T2SS of Example 3.2, the corresponding derived T1SS (F ₁, B) where B = { calorie rich diet , fibre rich diet , carbohydrate rich diet , soft diet , solid diet } is obtained as, ( F 1 , B ) = { ( calorie rich diet , { rice , noodles , ice cream } ) , ( fibre rich diet , { chicken sandwich , noodles , chicken soup } ) ( carbohydrate rich diet , { rice } ) , ( solid diet , { rice , noodles , chicken snadwich } ) ( soft diet , { pudding , ice cream , chicken soup } ) }

Remark 3.5. Although the above soft representation provides some information, it lacks detailing regarding the characteristics of food items such as whether a particular item is a “fibre rich, solid diet” or “carbohydrate rich solid diet” etc. In this respect, executing AND operation over the derived T1SS itself, adjoins some information but the associated process is quite laborious. Moreover, the resultant T1SS obtained would involve quite a considerable amount of irrelevant data which ultimately creates confusion in case of further computations. As for example, considering the derived T1SS in Example 3.4, (F,  B) ∧ (F, B) involves parameters such as “fibre rich diet, calorie rich diet” or “calorie rich diet, carbohydrate rich diet” etc. which are of no relevance to the actual T2SS.

Also, since in case of practical implementation of type-2 soft sets, the primary parameters and underlying parameters usually denote characteristics of different genre of the elements of the universe, resulting in a two-fold parameterization of data, T2SS are more general and hence a more efficient tool as compared to T1SS.

Definition 3.6 ( Absolute T2SS ).  A T2SS [ F , A ] is said to be an absolute T2SS if and only if for each parameter e ∈ A, the T1SS corresponding to F ( e ) is an absolute T1SS. An absolute T2SS is denoted by A ˜.

Definition 3.7 ( Null T2SS ).  A T2SS [ F , A ] is said to be a null T2SS if and only if for each parameter e ∈ A, the T1SS corresponding to F ( e ) is a null T1SS. A null T2SS is denoted by Φ ˜.

Remark 3.8.  Often while dealing with a decision making problem or a problem pertaining to practical importance, varying degrees of importance have to be assigned to the various elements of the universe. In our case this situation is taken care of by assigning different weights to the different elements of the parameter set under consideration. This notion has been materialized in the form of weighted type-2 soft sets.

Definition 3.9 ( Weighted T2SS ).  A T2SS is said to be a weighted type-2 soft set if both the primary and the underlying parameters have varying degrees of weights associated with them, where corresponding to a parameter p, the associated weight w is a real number such that 0 ≤ w ≤ 1.

4 Operations over type-2 soft sets

In this section we define some operations on T2SS and study some of their basic properties.

Definition 4.1 ( Containment ). A T2SS [ F , A ] is said to be a subset of a T2SS [ G , B ] if and only if,

A ⊂ B

In this case, [ F , A ] is said to be a soft subset of [ G , B ] and it is denoted by, [ F , A ] ⊏ [ G , B ] while [ G , B ] is said to be a soft superset of [ F , A ] and it is denoted by, [ G , B ] ⊐ [ F , A ].

Definition 4.2 ( Equality ). Two T2SS [ F , A ] and [ G , B ] are said to be equal, denoted as, [ F , A ] = [ G , B ] iff they are contained in each other.

Definition 4.3 ( Union ). The union of two T2SS [ F , A ] and [ G , B ] is a T2SS denoted by [ F , A ] ⊔ [ G , B ] = [ H , C ], where C = A ∪ B, is defined ∀e ∈ C, as H ( e ) = { F ( e ) , if e ∈ A - B G ( e ) , if e ∈ B - A F ( e ) ∪ ˜ G ( e ) , if e ∈ A ∩ B

It may be mentioned here that F ( e ) ∪ ˜ G ( e ) for all e ∈ A ∩ B, refers to the usual soft union between the respective T1SS corresponding to F ( e ) and G ( e ) respectively.

Definition 4.4 ( Intersection ). The intersection of two T2SS [ F , A ] and [ G , B ] is a T2SS denoted by [ F , A ] ⊓ [ G , B ] = [ K , C ] where C = A ∩ B, is defined by K ( e ) = F ( e ) ∩ ˜ G ( e ) , ∀ e ∈ C , where F ( e ) ∩ ˜ G ( e ) refers to the usual soft intersection between the respective T1SS corresponding to F ( e ) and G ( e ) respectively.

Example 4.5. Let X = {h ₁,  h ₂,  h ₃,  h ₄,  h ₅} be a set of five houses in a locality which are to be put on sale. Suppose the set of parameters be E = { beautiful , single storeyed , wooden , in good repari , spacious , near the market , in green surroundings , furnished , luxurious , with good security , with pool }

Also let, A = {beautiful,  luxurious} and B = {beautiful,  spacious} be two sets of parameters such that A,  B ⊂ E. Suppose two T2SS [ F , A ] and [ G , B ] be defined over X as, F ( beautiful ) = { wooden { h 2 , h 5 } , in green surroundings { h 1 , h 2 , h 3 , h 4 } } F ( luxurious ) = { wooden { h 5 } , with good security { h 1 , h 3 , h 5 } } G ( spacious ) = { wooden { h 5 } , with pool { h 3 , h 5 } } G ( beautiful ) = { wooden { h 2 , h 5 } , near the market { h 4 } , with good security { h 2 , h 3 } }

Then, [ F , A ] ⊔ [ G , B ] = [ H , C ], where C = A ∪ B, is defined ∀e ∈ A ∪ B, as, H ( beautiful ) = { in green surroundings { h 1 , h 2 , h 3 , h 4 } , wooden { h 2 , h 5 } , near the market { h 4 } , with good security { h 2 , h 3 } } H ( luxurious ) = { wooden { h 5 } , with good security { h 1 , h 3 , h 5 } } H ( spacious ) = { wooden { h 5 } , with pool { h 3 , h 5 } }

Also, [ F , A ] ⊓ [ G , B ] = [ K , C ], where C = A ∩ B, is defined ∀e ∈ A ∩ B, as, H ( beautiful ) = { wooden { h 2 , h 5 } }

Proposition 4.6. Let [ F , A ] , [ G , B ] and [ H , C ] be three T2SS defined over a common soft universe(X, E). Then, the following relations hold:

[ F , A ] ⊔ [ F , A ] = [ F , A ]

The proofs of the above propositions follow directly from the fact that union and intersection over T1SS are idempotent, commutative, associative and distributive.

Definition 4.7 ( AND Operation ). The AND operation between two T2SS [ F , A ] and [ G , B ] is denoted as, [ F , A ] ∧ [ G , B ] and is defined as [ F , A ] ∧ [ G , B ] = [ A , A × B ] , where A ( a , b ) = F ( a ) ∧ G ( b ) , ∀ ( a , b ) ∈ A × B .

Definition 4.8 ( OR Operation ). The OR operation between two T2SS [ F , A ] and [ G , B ] is denoted as, [ F , A ] ∨ [ G , B ] and is defined as [ F , A ] ∨ [ G , B ] = [ O , A × B ] , where O ( a , b ) = F ( a ) ∨ G ( b ) , ∀ ( a , b ) ∈ A × B .

Example 4.9.We consider the T2SS [ F , A ] and [ G , B ] of Example 4.5. Thus, we can obtain [ F , A ] ∧ [ G , B ] = [ A , A × B ], where

A ( a , b ) = F ( a ) ∧ G ( b ) , ∀ ( a , b ) ∈ A × B as, A ( beautiful , spacious ) = { wooden { h 5 } , wooden , with pool { h 5 } , in green surroungings , wooden φ , in green surroundings , with pool { h 3 } } A ( beautiful , beautiful ) = { wooden { h 2 , h 5 } , wooden , near the market φ , wooden , with good security { h 2 } , in green surroundings , wooden { h 2 } in green surroundings , near the market { h 4 } , in green surroundings , with good security { h 2 , h 3 } } A ( luxurious , spacious ) = { wooden { h 5 } , wooden , with pool { h 5 } , with good security , wooden { h 5 } , with good security , with pool { h 3 , h 5 } } A ( ( luxurious , beautiful ) = { wooden { h 5 } , wooden , near the market φ , wooden , with good security φ , with good security , wooden φ , with good security , near the market φ , with good security { h 3 } }

We also have, [ F , A ] ∨ [ G , B ] = [ O , A × B ], where O ( a , b ) = F ( a ) ∨ G ( b ) , ∀ ( a , b ) ∈ A × B, defined as, O ( beautiful , spacious ) = { wooden { h 2 , h 5 } , wooden , with pool { h 2 , h 3 , h 5 } , in green surroundings , wooden { h 1 , h 2 , h 3 , h 4 , h 5 } , in green surroundings , with pool { h 1 , h 2 , h 3 , h 4 , h 5 } } O ( ( beautiful , beautiful ) = { wooden { h 2 , h 5 } , wooden , near the market { h 2 , h 4 , h 5 } , wooden , with good security { h 2 , h 3 , h 5 } , in green surrounding , wooden { h 1 , h 2 , h 3 , h 4 , h 5 } , in green surrounding , near the market { h 1 , h 2 , h 3 , h 4 } , in green surrounding , with good security { h 1 , h 2 , h 3 , h 4 } } O ( luxurious , spacious ) = { wooden { h 5 } , wooden , with pool { h 3 , h 5 } with good security , wooden { h 1 , h 3 , h 5 } with good security , with pool { h 1 , h 3 , h 5 } } O ( luxurious , beautiful ) = { wooden { h 2 , h 5 } , wooden , near the market { h 4 , h 5 } , wooden , with good security { h 2 , h 3 , h 5 } with good security , wooden { h 1 , h 2 , h 3 , h 5 } , with good security , near the market { h 1 , h 3 , h 4 , h 5 } with good security { h 1 , h 2 , h 3 , h 5 } }

In order to explain this example, we consider the type-2 soft set [ A , A × B ], where the primary parameter “beautiful, spacious” literally denotes houses that are beautiful as well as spacious and includes the houses h ₅, which is wooden and has a pool as well and h ₃ which is a house in green surroundings with a pool. On the other hand, when the type-2 soft set [ O , A × B ] is considered, the parameter “beautiful, spacious” now denotes houses that may be beautiful or spacious and it includes wooden houses h ₂,  h ₅; wooden houses or houses with pool h ₂,  h ₃,  h ₅; wooden houses or houses in green surroundings h ₁,  h ₂,  h ₃,  h ₄,  h ₅ and houses in green surroundings or with pool h ₁,  h ₂,  h ₃,  h ₄,  h ₅.

Remark 4.10. In this case while performing AND or OR operations, if corresponding to a primary parameter, an underlying parameter is repeated in the corresponding T1SS, as for example, “wooden, wooden”, then it is simply put as “wooden”, just once without any repetition since doing so does not change any meaning nor does it change any characteristic of the corresponding T1SS, it rather simplifies the representation of the sets.

Proposition 4.11. Let [ F , A ] , [ G , B ] and [ H , C ] be three T2SS defined over a common soft universe (X, E). Then, the following relations hold:

[ F , A ] ∧ [ G , B ] = [ G , B ] ∧ [ F , A ]

The proofs of the above propositions are straight-forward and they follow from the fact that the AND and OR operations of the T1SS abide by the commutative, associative and distributive laws.

Definition 4.12 (Complement of a T2SS). The complement of a T2SS [ F , A ] is denoted as [ F , A ] c and is defined by [ F , A ] c = [ F c , A ] where F c ( α ) = ( F α , S α ) c , ∀ α ∈ A such that F α c ( β ) = X - F α ( β ) , ∀ β ∈ S α .

Example 4.13. We consider the T2SS [ F , A ] in Example 4.5. The complement of [ F , A ] is given by, F c ( beautiful ) = { wooden { h 1 , h 3 , h 4 } , in green surroundings { h 5 } } F c ( luxurious ) = { wooden { h 1 , h 2 , h 3 , h 4 } , with good security { h 2 , h 4 } }

The resultant T2SS may be interpreted as houses that cannot be classified as beautiful include the wooden houses h ₁,  h ₃,  h ₄ and the house h ₅ which is in green surroundings. Similar explanations apply for the rest of the parameters.

Proposition 4.14

( [ F , A ] c ) c = [ F , A ]

Proofs are straight-forward.

Proposition 4.15. Let [ F , A ] be T2SS over the soft universe (X, E) and A ˜ and Φ ˜ are the absolute and null T2SS respectively. Then,

Φ ˜ c = A ˜

Proofs are straight-forward.

5 Image of a T2SS under a mapping

Definition 5.1. Let X and Y be two non-empty sets and E be a set of parameters. Let [ F , A ] , A ⊂ E be a T2SS over (X, E). If f : X → Y be a mapping, then the image of [ F , A ] under the mapping f is defined by, f ( [ F , A ] ) = [ f ( F ) , A ] where [ f ( F ) ] ( α ) = f ( F α , S α ) , ∀ α ∈ A and [ f ( F α ) ] : S α → P ( Y )  is defined by [ f ( F α ) ] ( β ) = f ( F α ( β ) ) , ∀ β ∈ S α

Definition 5.2. Let X and Y be two non-empty sets and E be a set of parameters. Let [ G , B ] , B ⊂ E be a T2SS over (Y, E). If f : X → Y be an onto mapping, then the inverse image of [ G , B ] under the mapping f is defined by, f - 1 ( [ G , B ] ) = [ f - 1 ( G ) , B ] , where [ f - 1 ( G ) ] ( α ) = f - 1 ( G α , T α ) = ( f - 1 ( G α ) , T α ) , ∀ α ∈ B and [ f - 1 ( G α ) ] : T α → P ( X ) is defined by [ f - 1 ( G α ) ] ( β ) = f - 1 ( G α ( β ) ) , ∀ β ∈ T α .

Example 5.3. We refer to Example 4.5. Let X = {h ₁,  h ₂,  h ₃,  h ₄,  h ₅} be a set of five houses in a locality. Also let, Y = { h 1 ′ , h 2 ′ , h 3 ′ , … h 10 ′ } be a set of ten houses in another locality. Suppose, f : X → Y is a function that relates houses bearing similar characteristics in the two localities and is defined as, f ( h 1 ) = h 6 ′ ; f ( h 2 ) = h 3 ′ ; f ( h 3 ) = h 5 ′ ; f ( h 4 ) = h 7 ′ ; f ( h 5 ) = h 8 ′ Consider the T2SS as specified in Example 4.5.

The image of [ F , A ] under the mapping f is given by [ f ( F ) ] ( beautiful ) = { wooden { h 3 ′ , h 8 ′ } , in green surroundings { h 6 ′ , h 3 ′ , h 5 ′ , h 7 ′ } } [ f ( F ) ] ( luxurious ) = { wooden { h 8 ′ } , with good security { h 6 ′ , h 5 ′ , h 8 ′ } }

Proposition 5.4. Let X and Y be two non-empty setsand E be a set of parameters and [ F 1 , A ] and[ F 2 , A ] , A ⊂ E be two T2SS over (X, E). If f  :  X → Y be a mapping then it follows that:

[ F 1 , A ] ⊑ [ F 2 , A ] ⇒ f [ F 1 , A ] ⊑ f [ F 2 , A ]

Proof. Let for α ∈ A, F 1 ( α ) = ( F α 1 , S α 1 ) and F 2 ( α ) = ( F α 2 , S α 2 ). Then,

Suppose that [ F 1 , A ] ⊑ [ F 2 , A ].

Then ∀ α ∈ A , F 1 ( α ) ⊆ ˜ F 2 ( α )

i.e. ( F α 1 , S α 1 ) ⊆ ˜ ( F α 2 , S α 2 ).

Now, [ f ( F 1 ) ( α ) ] = f ( F α 1 , S α 1 ) = ( f ( F α 1 ) , S α 1 ) , ∀ α ∈ A such that ∀ β ∈ S α 1 [ f ( F α 1 ) ] ( β ) = f ( F α 1 ( β ) ) ⊆ f ( F α 2 ( β ) ) = [ f ( F α 2 ) ] ( β ) and hence the proof follows.

The rest of the proofs follow in a similar way.

Proposition 5.5. Let X and Y be two non-empty sets and E be a set of parameters and [ G 1 , B ] and [ G 2 , B ] , B ⊂ E be two T2SS over (Y, E). If f : X → Y be an onto mapping then it follows that:

[ G 1 , B ] ⊑ [ G 2 , B ] ⇒ f - 1 [ G 1 , B ] ⊑ f - 1 [ G 2 , B ]

Proof. Let for α ∈ B,

G 1 ( α ) = ( G α 1 , T α 1 ) and G 2 ( α ) = ( G α 2 , T α 2 ).

(i) Suppose that, [ G 1 , B ] ⊑ [ G 2 , B ]. Now, [ f - 1 ( G 1 ) ] ( α ) = f - 1 ( G α 1 , T α 1 ) = ( f - 1 ( G α 1 ) , T α 1 ) , ∀ α ∈ B where ∀ β ∈ T α 1, [ f - 1 ( G α 1 ) ] ( β ) = f - 1 ( G α 1 ( β ) ) ⊆ f ( G α 2 ( β ) ) = [ f ( G α 2 ) ] ( β ) .

Thus, [ G 1 , B ] ⊑ [ G 2 , B ] ⇒ f - 1 [ G 1 , β ] ⊑ f - 1 [ G 2 , B ] The rest of the proofs follow in a similar way.

Proposition 5.6. Let X and Y be two non-empty sets and E be a set of parameters and [ F , A ] and [ G , B ] , A , B ⊂ E be arbitrary T2SS over (X, E) and (Y, E) respectively. If f : X → Y be a mapping then,

[ F , A ] ⊑ f - 1 { f [ F , A ] }, equality holds when f is injective.