arXiv:1307.4960v1 [math.GM] 17 Jul 2013
Convex and Concave Soft Sets and Some Properties Irfan Deli Department of Mathematics, Faculty of Arts and Sciences, Kilis 7 Aralık University, 79000 Kilis, Turkey,
[email protected] July 19, 2013 Abstract In this study, after given the definition of soft sets and their basic operations we define convex soft sets which is an important concept for operation research, optimization and related problems. Then, we define concave soft sets and give some properties for the concave sets. For these, we will use definition and properties of convex-concave fuzzy sets in literature. We also give different some properties for the convex and concave soft sets. Keyword 0.1 Soft sets, fuzzy set, α-inclusion, Convex soft sets, concave soft sets.
1
Introduction
Many fields deal with the uncertain data which may not be successfully modeled by the classical mathematics, probability theory, fuzzy sets [33], rough sets [28], and other mathematical tools. In 1999, Molodtsov [26] proposed a completely new approach so-called soft set theory that is more universal for modeling vagueness and uncertainty. After definitions of the operations of soft sets [3, 10, 25, 30], the properties and applications on the soft set theory have been studied increasingly (e.g. [3, 8, 9, 13, 14]). The algebraic structure of soft set theory has also been studied increasingly (e.g. [1, 2, 18, 19]). In recent years, many interesting applications of soft set theory have been expanded by embedding the ideas of fuzzy sets (e.g. [4, 11, 12, 15, 19, 21, 24]), rough sets (e.g. [4, 16, 17]) and intuitionistic fuzzy sets (e.g. [20, 22, 23, 27]). Convex (concave) and convex (concave )fuzzy sets play important roles in optimization theory. Various definitions of convex fuzzy and concave fuzzy sets have appeared in the literature. A significant definition of convex fuzzy sets 1
introduced by Zadeh [33] and concave fuzzy sets introduced by Chaudhuri [6]. Then, concavoconvex fuzzy sets proposed by Sarkar [29], which is convex and concave fuzzy sets together conceived by combining. Moreover, some properties of the concavoconvex fuzzy set are given by the author. Subsequently, works on convex (concave )fuzzy sets in theories and applications has been progressing rapidly by [5, 31, 32]. In this work, we introduce the soft version of the Zadeh’s definition of fuzzy convex set, and we will call this generalization an soft convex sets. Also, we will introduce the definition of soft concave set and we study some desired properties. The plan of the paper is as follows. In Section 2, we give some notations, definitions used throughout the paper. In Section 3, we define soft convex sets and soft concave sets and then we show some properties.
2
Preliminary
In this section, we present the basic definitions and results of soft set theory [10]. More detailed explanations related to this subsection may be found in [10, 25, 26]. Definition 2.1 [26] Let U be a universe, P (U ) be the power set of U and E be a set of parameters that are describe the elements of U . A soft set S over U is a set defined by a set valued function S representing a mapping fS : E → P (U ) It is noting that the soft set is a parametrized family of subsets of the set U , and therefore it can be written a set of ordered pairs S = {(x, fS (x)) : x ∈ E} Here, fS is called approximate function of the soft set S and fS (x) is called x-approximate value of x ∈ E. The subscript S in the fS indicates that fS is the approximate function of S. Generally, fS , fT , fV , ... will be used as an approximate functions of S, T , V , ..., respectively. Note that if fS (x) = ∅, then the element (x, fS (x)) is not appeared in S. Example 2.2 Suppose that U = {u1 , u2 , u3 , u4 , u5 , u6 } is the universe and E = {x1 , x2 , x3 , x4 , x5 } is the set of parameters. If fS (x1 ) = {u1 , u2 , u3 }, fS (x2 ) = {u1 , u4 }, fS (x3 ) = ∅, fS (x4 ) = U then the soft-set S is written by S = {(x1 , {u1 , u2 , u3 }), (x2 , {u1 , u4 }), (x4 , U )} By using same parameter set E, we can construct a soft set T different then S. Assume that fT (x1 ) = {u1 , u2 , u6 }, fT (x2 ) = {u1 , u2 , u3 }, fT (x3 ) = {u1 , u2 }, fT (x4 ) = ∅, fT (x5 ) = {u6 } then the soft-set T is written by T = {(x1 , {u1 , u2 , u6 }), (x2 , {u1 , u2 , u3 }), (x3 , {u1 , u2 }), (x5 , {u6 })} 2
Definition 2.3 [10] Let S and T be two soft sets. Then, a) If fS (x) = ∅ for all x ∈ E, then S is called a empty soft set, denoted by SΦ . ˜ . b) If fS (x) ⊆ fT (x) for all x ∈ E, then S is a soft subset of T , denoted by S ⊆T Definition 2.4 [10] Let S and T be two soft sets. Then, a) Complement of S is denoted by S c˜. Its approximate function fS c˜ is defined by fS c˜ (x) = U \ fS (x) for all x ∈ E ˜ T . Its approximate function fS ∪eT is b) Union of S and T is denoted by S ∪ defined by fS ∪e T (x) = fS (x) ∪ fT (x) for all x ∈ E. ˜ T . Its approximate function fS ∩e T c) Intersection of S and T is denoted by S ∩ is defined by fS ∩e T (x) = fS (x) ∩ fT (x) for all x ∈ E. Definition 2.5 [7] Let S be a soft set over U and α be a subset of U. Then, α-inclusion of the soft set S, denoted by S α , is defined as S α = {x ∈ E : fS (x) ⊇ α}
3
Convex and concave soft sets
In this section, we define convex soft sets and concave soft sets and then give desired some properties. Some of it is quoted from example in [5, 31, 29, 32, 33]. In this paper, E will denote the n-dimensional Euclidean space Rn and U denotes the arbitrary set. Definition 3.1 The soft set S on E is called a convex soft set if fS (ax + (1 − a)y) ⊇ fS (x) ∩ fS (y) for every x, y ∈ E and a ∈ I. Theorem 3.2 S ∩ T is a convex soft set when both S and T are convex soft sets. Proof: Suppose that there exist x, y ∈ E and a ∈ I and W = S ∩ T . Then, fW (ax + (1 − a)y) = fS (ax + (1 − a)y) ∩ fT (ax + (1 − a)y)
(1)
Now, since S and T convex, fS (ax + (1 − a)y) ⊇ fS (x) ∩ fS (y)
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(2)
fT (ax + (1 − a)y) ⊇ fT (x) ∩ fT (y)
(3)
fW (ax + (1 − a)y) ⊇ (fS (x) ∩ fS (y)) ∩ (fT (x) ∩ fT (y))
(4)
fW (ax + (1 − a)y) ⊇ fW (x) ∩ fW (y)
(5)
and hence,
and thus
Remark 3.3 If {Si : i ∈ {1, 2, ...}} is any family of convex soft sets, then the intersection ∩i∈I Si is a convex soft set. Theorem 3.4 The union of any family {Si : i ∈ I = {1, 2, ...}} of convex soft sets is not necessarily a convex soft set. Proof:The proof is straightforward. Theorem 3.5 S is a convex soft set on E iff for every β ∈ [0, 1] and α ∈ P (U ), S α is a convex set on E. Proof:⇒ Assume that S is a convex soft set. If x1 , x2 ∈ E and α ∈ P (U ), then fS (x1 ) ⊇ α and fS (x2 ) ⊇ α. It follows from the convexity of S that fS (βx1 + (1 − β)x2 ) ⊇ fS (x1 ) ∩ fS (x2 ) and thus S α is a convex set. ⇐ Assume that S α is a convex set for every β ∈ [0, 1]. Especially, for x1 , x2 ∈ E, S α is convex for α = fS (x1 ) ∩ fS (x2 ). Since fS (x1 ) ⊇ α and fS (x2 ) ⊇ α, we have x1 ∈ S α and x2 ∈ S α , whence βx1 + (1 − β)x2 ∈ S α . Therefore, fS (βx1 + (1 − β)x2 ) ⊇ α = fS (x1 ) ∩ fS (x2 ), which indicates S is a convex soft set on X. Definition 3.6 The soft set S on E is called a concave soft set if fS (ax + (1 − a)y) ⊆ fS (x) ∪ fS (y) for every x, y ∈ E and a ∈ I. Theorem 3.7 S ∪ T is a concave soft set when both S and T are concave soft sets. Proof: Suppose that there exist x, y ∈ E and a ∈ I and W = S ∪ T . Then, fW (ax + (1 − a)y) = fS (ax + (1 − a)y) ∪ fT (ax + (1 − a)y)
(6)
Now, since S and T concave, fS (ax + (1 − a)y) ⊆ fS (x) ∪ fS (y) 4
(7)
fT (ax + (1 − a)y) ⊆ fT (x) ∪ fT (y)
(8)
fW (ax + (1 − a)y) ⊆ (fS (x) ∪ fS (y)) ∪ (fT (x) ∪ fT (y))
(9)
fW (ax + (1 − a)y) ⊆ fW (x) ∪ fW (y)
(10)
and hence,
and thus
Remark 3.8 If {Si : i ∈ {1, 2, ...}} is any family of concave soft sets, then the union ∪i∈I Si is a concave soft set. Theorem 3.9 S ∩ T is a concave soft set when both S and T are concave soft sets. Proof: Suppose that there exist x, y ∈ E and a ∈ I and W = S ∩ T . Then, fW (ax + (1 − a)y) = fS (ax + (1 − a)y) ∩ fT (ax + (1 − a)y)
(11)
Now, since S and T concave, fS (ax + (1 − a)y) ⊆ fS (x) ∪ fS (y)
(12)
fT (ax + (1 − a)y) ⊆ fT (x) ∪ fT (y)
(13)
fW (ax + (1 − a)y) ⊆ (fS (x) ∪ fS (y)) ∩ (fT (x) ∪ fT (y))
(14)
fW (ax + (1 − a)y) ⊆ fW (x) ∩ fW (y)
(15)
and hence,
and thus
Remark 3.10 The intersection of any family {Si : i ∈ I = {1, 2, ...}} of concave soft sets is concave soft set. Theorem 3.11 S c is a concave soft set when S is a convex soft sets. Proof: Suppose that there exist x, y ∈ E, a ∈ I and S be a convex soft set. Then, since S is convex, fS (ax + (1 − a)y) ⊇ fS (x) ∩ fS (y)
(16)
U \ fS (ax + (1 − a)y) ⊆ U \ {fS (x) ∩ fS (y)}
(17)
or If fS (x) ⊃ fS (y) then we may write U \ fS (ax + (1 − a)y) ⊆ U \ fS (y). 5
(18)
If fS (x) ⊂ fS (y) then we may write U \ fS (ax + (1 − a)y) ⊆ U \ fS (x).
(19)
From (18) and (18), we have U \ fS (ax + (1 − a)y) ⊆ {U \ fS (x) ∪ U \ fS (y)}.
(20)
So, S c is a concave soft set. Theorem 3.12 S c is a convex soft set when S is a concave soft sets. Proof: Suppose that there exist x, y ∈ E, a ∈ I and S be a concave soft set. Then, since S is concave, fS (ax + (1 − a)y) ⊆ fS (x) ∪ fS (y)
(21)
U \ fS (ax + (1 − a)y) ⊇ U \ {fS (x) ∪ fS (y)}
(22)
or If fS (x) ⊃ fS (y) then we may write U \ fS (ax + (1 − a)y) ⊇ U \ fS (x).
(23)
If fS (x) ⊂ fS (y) then we may write U \ fS (ax + (1 − a)y) ⊇ U \ fS (y).
(24)
From (23) and (24), we have U \ fS (ax + (1 − a)y) ⊇ {U \ fS (x) ∩ U \ fS (y)}.
(25)
So, S c is a convex soft set. Theorem 3.13 S is a concave soft set on E iff for every β ∈ [0, 1] and α ∈ P (U ), S α is a concave set on E. Proof: ⇒ Assume that S is a concave soft set. If x1 , x2 ∈ E and α ∈ P (U ), then fS (x1 ) ⊇ α and fS (x2 ) ⊇ α. It follows from the concavity of S that fS (βx1 + (1 − β)x2 ) ⊆ fS (x1 ) ∪ fS (x2 ) and thus S α is a concave set. ⇐ Assume that S α is a concave set for every β ∈ [0, 1]. Especially, for x1 , x2 ∈ E, S α is concave for α = fS (x1 ) ∪ fS (x2 ). Since fS (x1 ) ⊇ α and fS (x2 ) ⊇ α, we have x1 ∈ S α and x2 ∈ S α , whence βx1 + (1 − β)x2 ∈ S α . Therefore, fS (βx1 + (1 − β)x2 ) ⊆ α = fS (x1 ) ∪ fS (x2 ), which indicates S is a concave soft set on X.
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4
Conclusion
In the literature, convex fuzzy sets has been introduced widely by many researchers. In this paper, we defined convex soft sets and concave soft soft sets and give some properties. It can extend to strictly convex and strongly convex soft set by using strictly convex and strongly convex set which are useful in fuzzy optimization. It is worth pointing out that the results obtained here for convex soft sets can be generalized to strictly convex and strongly convex soft set by a similar way. Also we will try to explore characterizations of convex soft sets to optimization in the future. The theory may be applied to many fields and more comprehensive in the future to solve the related problems, such as; pattern classification, operation research, decision making, optimization problem, and so on.
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