/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 9, No. 3, 2016, 333-339 ISSN 1307-5543 – www.ejpam.com On Some New Operations In Probabilistic Soft Set Theory Çiğdem Gunduz(Aras)∗, Hande Poşul Department of Mathematics, Kocaeli University,Kocaeli 41380, Turkey Abstract. In this paper, we study the theory of probabilistic soft sets introduced by [7]. We define equality of two probabilistic soft sets, subset, complement of a probabilistic soft set with examples. We also introduce the operations of union, intersection, difference and symmetric difference. We prove that certain De Morgan’s laws hold in probabilistic soft set theory with respect to these new definitions. 2010 Mathematics Subject Classifications: 03B52 Key Words and Phrases: Soft sets, Probabilistic soft sets 1. Introduction In theory, for formal modeling, reasoning, and computing we have traditional tools such as crisp, deterministic, and precise in character but in practical way we see that data in eco- nomics, engineering, environment, social science, medical science, etc. are not always all crisp and classical methods because of various types of uncertainties present in these problems can not be used, successfully. There are some theories like theory of probability, theory of fuzzy sets and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. According to Molodtsov [5], since all these theories have their inherent difficulties the concept of soft set theory as a mathematical tool for dealing with uncertainties which is free from the above difficulties has been initiated in [5]. Soft set theory has a rich potential for applications in several directions [1–4, 6]. Zhu and Wen [7] have proposed the notion of probabilistic soft sets incorporated Molodtsov’s soft set theory with probability the- ory and introduced three operations with probabilistic soft sets the conditional probabilistic soft set. In the present paper, we make a theoretical study of the "Probabilistic soft set theory" in more detail. ∗Corresponding author. Email address: caras@kocaeli.edu.tr (Ç. Gunduz(Aras)) http://www.ejpam.com 333 c© 2016 EJPAM All rights reserved. Ç. Gunduz(Aras), H. Poşul / Eur. J. Pure Appl. Math, 9 (2016), 333-339 334 2. Preliminary Definition 1. Let U be a universe. A probabilistic set X over U is a set defined by a function µX representing a mapping µX : U → I = [0,1] satisfying the following conditions: (i) For each ∼ U ⊂ U, ∑ u∈ ∼ U µX (u)≤ 1 (ii) If ∼ U = U, then ∑ u∈ ∼ U µX (u) = 1 or ∑ u∈ ∼ U µX (u) = 0 µX is called the the probabilistic membership function of X , and the value µX (u) is called the probabilistic grade of membership of u ∈ U. Thus a probabilistic set X over U can be represented as follows: X = { � µX (u)/u � : u ∈ U}. Note that the set of all the probabilistic sets over U will be denoted by Pr (U). Example 1. Let U = {u1,u2,u3,u4} be a universal set. Then X = { � 0.4/u1 � , � 0.1/u2 � , � 0.2/u3 � , � 0.3/u4 � } is a probabilistic set over U. Definition 2. A probabilistic set X over U is called empty probabilistic set if its membership function is zero everywhere in U and denoted by ;. i.e, µX : U → I ,µX (u) = 0. Example 2. Let U = {u1,u2,u3,u4,u5} is a universal set. Then X = �� 0/u1 � , � 0/u2 � , � 0/u3 � , � 0/u4 � , � 0/u5 � = ; is an empty probabilistic set. 3. Probabilistic Soft Set In this section, we define probabilistic soft sets and their operations. From now on, we will use ΓP A ,ΓP B , . . .. etc, for probabilistic soft sets and γP A ,γP B , . . .. etc. for their probabilistic approximate functions, respectively. Throughout this work, U refers to an initial universe, E is a set of parameters and A⊂ E. Definition 3. A probabilistic soft set (prs-set) ΓP A over U is a set defined by a function γP A repre- senting a mapping γP A : E→ Pr (U) such that γP A (x) = ; if x /∈ A. Ç. Gunduz(Aras), H. Poşul / Eur. J. Pure Appl. Math, 9 (2016), 333-339 335 Here, γP A is called probabilistic approximate function of the probabilistic soft set ΓP A . Hence prob- abilistic soft set ΓP A over U can be represented by the set of ordered pairs ΓP A = { � x ,γP A (x) � : x ∈ E,γP A (x) ∈ Pr (U)}. Note that the set of all probabilistic soft set ΓP A over U will be denoted by Pr S (U). Example 3. Assume that U = {u1,u2,u3,u4,u5} is a universal set and E = {x1, x2, x3, x4} is a set of all parameters. If A= {x1, x3, x4}, γP A � x1 � ={0.9/u2, 0.1/u4} γP A � x3 � ={0.2/u1, 0.2/u2, 0.2/u3, 0.2/u4, 0.2/u5} γP A � x4 � ={0.2/u1, 0.4/u3, 0.4/u5} then the prs-set ΓP A is written ΓP A ={ � x1, {0.9/u2, 0.1/u4} � , � x3, {0.2/u1, 0.2/u2, 0.2/u3, 0.2/u4, 0.2/u5} � , � x4, {0.2/u1, 0.4/u3, 0.4/u5} � }. Definition 4. Let ΓP A ∈ Pr S (U). If γP A (x) = ; for all x ∈ A then ΓP A is called A−impossible prs-set, denoted by ΓP Φ. Example 4. Assume that U = {u1,u2,u3,u4,u5} is a universal set and E = {x1, x2, x3, x4} is a set of all parameters. If A= {x1, x2}, and γP A � x1 � = ;, γP A � x2 � = ;, then probabilistic soft set ΓP A is an impossible prs-set, i.e. ΓP A = Γ P Φ. Definition 5. Let ΓP A ,ΓP B ∈ Pr S (U). Then ΓP A is a prs-subset of ΓP B , denoted by ΓP A e⊆ΓP B , if A ⊂ B and γP A (x) ⊆ γ P B (x) for all x ∈ A. Remark 1. As in the definition of the classical subset, ΓP A e⊆ΓP B does not imply that every element of ΓP A is an element of ΓP B . Example 5. Assume that U = {u1,u2,u3,u4,u5} is a universal set and E = {x1, x2, x3} is a set of all parameters. x1→{0.4/u1, 0.2/u2, 0.4/u5} x2→{0.2/u1, 0.4/u2, 0.1/u3, 0.1/u4, 0.2/u5} x3→{1/u3} If A= {x1}, B = {x1, x2}, then γP A � x1 � ={0.4/u1, 0.2/u2} γP B � x1 � ={0.4/u1, 0.2/u2, 0.4/u5} Ç. Gunduz(Aras), H. Poşul / Eur. J. Pure Appl. Math, 9 (2016), 333-339 336 γP B � x2 � ={0.2/u1, 0.4/u2, 0.1/u3}. Hence ΓP A ={ � x1, {0.4/u1, 0.2/u2} � } ΓP B ={ � x1, {0.4/u1, 0.2/u2, 0.4/u5} � , � x2, {0.2/u1, 0.4/u2, 0.1/u3} � } Then for all x ∈ E, γP A (x) ⊆ γ P B (x), hence ΓP A e⊆ΓP B . But it is clear that � x1, {0.4/u1, 0.2/u2} � ∈ ΓP A , but � x1, {0.4/u1, 0.2/u2} � /∈ ΓP B . Proposition 1. Let ΓP A ,ΓP B ∈ Pr S (U). Then (i) ΓP A e⊆ΓP A (ii) ΓP A e⊆ΓP B and ΓP B e⊆ΓP C ⇒ Γ P A e⊆ΓP C . Proof. They can be proved easily by using the probabilistic approximate function of prs- set. Definition 6. Let ΓP A ,ΓP B ∈ Pr S (U). Then ΓP A and ΓP B are prs-equal set written as ΓP A = Γ P B , if ΓP A is a prs-subset of ΓP B and ΓP B is a prs-subset of ΓP A . Proposition 2. Let ΓP A ,ΓP B ,ΓP C ∈ Pr S (U). Then (i) ΓP A = Γ P B and ΓP B = Γ P C ⇒ Γ P A = Γ P C (ii) ΓP A e⊆ΓP B and ΓP B e⊆ΓP A⇔ ΓP A = Γ P B . Proof. The proofs are straightforward. Definition 7. Let ΓP A ,ΓP B ∈ Pr S (U). Then the difference of ΓP A and ΓP B , denoted by ΓP A e\ΓP B , is defined by its probabilistic approximate functions: γP A\B (x) = γ P A (x) \ γ P B (x) , for allx ∈ E. Definition 8. Let ΓP A ,ΓP B ∈ Pr S (U) and ΓP A e⊆ΓP B . Then the complement of ΓP A on ΓP B , denoted by� ΓP A �c ΓP B , is defined by � γP A �c γP B (x) = γP B (x) \ γ P A (x) , for all x ∈ E. Example 6. Let us consider Example 5. Then, � ΓP A �c ΓP B = { � x1, {0.4/u5} � , � x2, {0.2/u1, 0.4/u2, 0.1/u3 � }. Definition 9. Let ΓP A ,ΓP B ∈ Pr S (U). Then the union of ΓP A and ΓP B , denoted by ΓP A e∪ΓP B , is defined by its probabilistic approximate functions: γP A∪B (x) = γ P A (x)∪ γ P B (x) , for all x ∈ E. Ç. Gunduz(Aras), H. Poşul / Eur. J. Pure Appl. Math, 9 (2016), 333-339 337 Example 7. Assume that U = {u1,u2,u3,u4,u5} is a universal set and E = {x1, x2, x3, x4} is a set of all parameters. x1→{0.4/u1, 0.2/u2, 0.4/u5} x2→{0.2/u1, 0.4/u2, 0.1/u3, 0.1/u4, 0.2/u5} x3→{1/u3} x4→{0.2/u1, 0.1/u2, 0.3/u3, 0.2/u4, 0.2/u5} If A= {x1, x2}, B = {x1, x2, x4}, then γP A � x1 � ={0.2/u2, 0.4/u5} γP A � x2 � ={0.2/u1, 0.1/u4} γP B � x1 � ={0.4/u1, 0.2/u2, 0.4/u5} γP B � x2 � ={0.4/u2, 0.1/u3} γP B � x4 � ={0.2/u4, 0.2/u5} Hence ΓP A ={ � x1, {0.2/u2, 0.4/u5} � , � x2, {0.2/u1, 0.1/u4} � } ΓP B ={ � x1, {0.4/u1, 0.2/u2, 0.4/u5} � , � x2, {0.4/u2, 0.1/u3} � , � x4, {0.2/u4, 0.2/u5} � } It is clear that A∪ B = {x1, x2, x4} and γP A∪B � x1 � =γP A � x1 � ∪ γP B � x1 � = {0.4/u1, 0.2/u2, 0.4/u5} γP A∪B � x2 � =γP A � x2 � ∪ γP B � x2 � = {0.2/u1, 0.4/u2, 0.1/u3, 0.1/u4} γP A∪B � x4 � =γP A � x4 � ∪ γP B � x4 � = {0.2/u4, 0.2/u5} i.e. ΓP A e∪ΓP B = { � x1, {0.4/u1, 0.2/u2, 0.4/u5} � , � x2, {0.2/u1, 0.4/u2, 0.1/u3, 0.1/u4} � � x4, {0.2/u4, 0.2/u5} � }. Proposition 3. Let ΓP A ,ΓP B ,ΓP C ∈ Pr S (U). Then (i) ΓP A e∪ΓP A = Γ P A (ii) ΓP A e∪ΓP B = Γ P B e∪ΓP A (iii) � ΓP A e∪ΓP B � e∪ΓP C = Γ P A e∪ � ΓP B e∪ΓP C � . Proof. The proofs can be proved easily by using the Definition 9. Definition 10. Let ΓP A ,ΓP B ∈ Pr S (U). Then the intersection of ΓP A and ΓP B , denoted by ΓP A e∩ΓP B , is defined by its probabilistic approximate functions: γP A∩B (x) = γ P A (x)∩ γ P B (x) , for all x ∈ A∩ B, A∩ B 6= ;. Ç. Gunduz(Aras), H. Poşul / Eur. J. Pure Appl. Math, 9 (2016), 333-339 338 Example 8. Let us consider Example 7. Then A∩ B = {x1, x2} and γP A∩B � x1 � =γP A � x1 � ∩ γP B � x1 � = {0.2/u2, 0.4/u5} γP A∩B � x2 � =γP A � x2 � ∩ γP B � x2 � = ; Hence ΓP A e∩ΓP B = { � x1, {0.2/u2, 0.4/u5} � } is obtained. Proposition 4. Let ΓP A ,ΓP B ,ΓP C ∈ Pr S (U). Then (i) ΓP A e∩ΓP A = Γ P A (ii) ΓP A e∩ΓP B = Γ P B e∩ΓP A (iii) � ΓP A e∩ΓP B � e∩ΓP C = Γ P A e∩ � ΓP B e∩ΓP C � . Proof. The proofs can be proved easily by using the Definition 10. Proposition 5. Let ΓP A ,ΓP B ,ΓP C ∈ Pr S (U) and ΓP A ,ΓP B e⊆ΓP C . Then, De Morgan’s laws for ΓP A ,ΓP B are valid as follows: (i) � ΓP A e∩ΓP A �c ΓP C = � ΓP A �c ΓP C e∪ � ΓP B �c ΓP C (ii) � ΓP A e∪ΓP B �c ΓP C = � ΓP A �c ΓP C e∩ � ΓP B �c ΓP C Proof. The proofs can be proved easily by using the respective probabilistic approximate functions. So, we only prove (i) case. For all x ∈ E, � γP A∩B �c γP C (x) = � γP A ∩ γ P B �c γP C (x) = γP C (x) \ � γP A ∩ γ P B � (x) = � γP C (x) \ γ P A (x) � ∪ � γP C (x) \ γ P B (x) � = � γP A �c γP C (x)∪ � γP B �c γP C (x) . Definition 11. Let ΓP A ,ΓP B ∈ Pr S (U). Then the symmetric difference of ΓP A and ΓP B , denoted by ΓP A e∆ΓP B , is defined by its probabilistic approximate functions: γP A (x)∆γ P B (x) = � γP A (x) \ γ P B (x) � ∪ � γP B (x) \ γ P A (x) � , for all x ∈ E. Example 9. Let us consider Example 7. Then ΓP A e∆ΓP B = �� x1, {0.4/u1} � , � x2, {0.2/u1, 0.4/u2, 0.1/u3, 0.1/u4} � � x4, {0.2/u4, 0.2/u5} � . is obtained. REFERENCES 339 Proposition 6. Let ΓP A ,ΓP B ,ΓP C ∈ Pr S (U). The following conditions are satisfied: (i) ΓP A e∆ΓP B = Γ P B e∆ΓP A (ii) � ΓP A e∆ΓP B � e∆ΓP C = Γ P A e∆ � ΓP B e∆ΓP C � (iii) ΓP A = Γ P B⇔ ΓP A e∆ΓP B = Γ P Φ. Proof. The proofs can be proved easily by using the respective probabilistic approximate functions. So, we only prove (i) case. For all x ∈ E, γP A (x)∆γ P B (x) = � γP A (x) \ γ P B (x) � ∪ � γP B (x) \ γ P A (x) � = � γP B (x) \ γ P A (x) � ∪ � γP A (x) \ γ P B (x) � =γP B (x)∆γ P A (x) i.e., ΓP A e∆ΓP B = Γ P B e∆ΓP A is obtained. 4. Conclusion In this paper, we study the theory of probabilistic soft sets. We give some operations such as union, intersection, difference and symmetric difference. We prove that certain De Morgan’s laws hold in probabilistic soft set theory with respect to these new definitions. 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