On Entropy and Similarity Measure of Interval Valued Neutrosophic Sets

Neutrosophic Sets and Systems

Volume 9 Article 8

1-1-2015

Ali Aydogdu

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On Entropy and Similarity Measure of Interval Valued Neutrosophic Sets

Ali Aydoğdu

Department of Mathematics, Atatürk University, Erzurum, 25400, Turkey. E-mail: [email protected]

Abstract. In this study, we generalize the similarity neutrosophic sets which generalizes the entropy measures measure of intuitionistic fuzzy set, which was defined by defined Wei, Wang and Zhang, for interval valued intui- Hung and Yang, to interval valued neutrosophic sets. tionistic fuzzy sets. Then we propose an entropy measure for interval valued

Keywords: Interval valued neutrosophic set; Entropy; Similarity measure.

1 Introduction has been given in section 4. In section 5 presents our conclusion. Neutrosophy is a branch of philosophy which studies the origin, nature and scope of neutralities. Smarandache [6] 2 Preliminaries introduced neutrosophic set by adding an inde-terminancy membership on the basis of intuitionistic fuzzy set. In this section, we give some basic definition re-lated Neutrosophic set generalizes the concept of the classic set, interval valued neutrosophic sets (IVNS) from [8]. fuzzy set [12], interval valued fuzzy set [7], intuitionistic Definition 2.1. Let 푋 be a universal set, with generic fuzzy set [1], interval valued intuitionistic fuzzy sets [2], element of 푋 denoted by 푥. An interval valued neutrosoph- etc. A neutrosophic set consider truth-membership, in- ic set (IVNS) 퐴 in 푋 is characterized by a truth- determinacy-membership and falsity-membership which membership function 푇퐴, indeterminacy-membership func- are completely independent. Wang et al. [9] introduced tion 퐼퐴 and falsity-membership function 퐹퐴, with for each single valued neutrosophic sets (SVNS) which is an in- 푥 ∈ 푋, 푇퐴(푥), 퐼퐴(푥), 퐹퐴(푥) ⊆ [0,1]. stance of the neutrosophic set. However, in many applica- When the universal set 푋 is continuous, an IVNS 퐴 can tions, the decision information may be provided with inter- be written as vals, instead of real numbers. Thus interval neutrosophic 〈푇 (푥), 퐼 (푥), 퐹 (푥)〉 퐴 = ∫ 퐴 퐴 퐴 , 푥 ∈ 푋. sets, as a useful generation of neutrosophic set, was intro- 푥 duced by Wang et al. [8]. Interval neutrosophic set de- 푋 When the universal set 푋 is discrete, an IVNS 퐴 can be scribed by a truth membership interval, an indeterminacy written as membership interval and false membership interval. 푛 Many methods have been introduced for measuring the 퐴 = ∑ 〈푇퐴(푥푖), 퐼퐴(푥푖), 퐹퐴(푥푖)〉/푥푖 , 푥 ∈ 푋. degree of similarity between neutrosophic set. Broumi and 푖=1 Smarandache [3] gave some similarity measures of neutro- Definition 2.2. An IVNS 퐴 is empty if and only if sophic sets. Ye [11] introduced similarity measures based inf 푇퐴(푥) = sup 푇퐴(푥) = 0, inf 퐼퐴(푥) = sup 퐼퐴(푥) = 1 and on the distances of interval neutrosophic sets. Further en- inf 퐹퐴(푥) = sup 퐹퐴(푥) = 0, for all 푥 ∈ 푋. tropy describes the degree of fuzziness in fuzzy set. Simi- Definition 2.3. An IVNS 퐴 is contained in the other larity measures and entropy of single valued neutrosophic IVNS 퐵, 퐴 ⊆ 퐵, if and only if sets was introduced by Majumder and Samanta [5] for the inf 푇퐴(푥) ≤ inf 푇퐵(푥) , sup 푇퐴(푥) ≤ sup 푇퐵(푥) first time. inf 퐼퐴(푥) ≥ inf 퐼퐵(푥) , sup 퐼퐴(푥) ≥ sup 퐼퐵(푥) The rest of paper is organized as it follows. Some prelim- inf 퐹퐴(푥) ≥ inf 퐹퐵(푥) , sup 퐹퐴(푥) ≥ sup 퐹퐵(푥) inary definitions and notations interval neutrosophic sets in for all 푥 ∈ 푋. the following section. In section 3, similarity measure be- Definition 2.4. The complement of an IVNS 퐴 is de- tween the two interval neutrosophic sets has been intro- noted by 퐴푐 and is defined by duced. The notation entropy of interval neutrosophic sets

Ali Aydoğdu, On Entropy and Similarity Measure of Interval Valued Neutrosophic Sets Neutrosophic Sets and Systems, Vol. 9, 2015 48

푇퐴푐(푥) = 퐹퐴(푥) We shall prove this similarity measure satisfies inf 퐼퐴푐(푥) = 1 − sup 퐼퐴(푥) the properties of the Definition 3.1. sup 퐼퐴푐(푥) = 1 − inf 퐼퐴(푥) Proof: We show that 푆(퐴, 퐵) satisfies all properties 1- 퐹퐴푐(푥) = 푇퐴(푥) 4 as above. It is obvious, the properties 1-3 is satisfied of for all 푥 ∈ 푋. definition 3.1. In the following we only prove 4. Definition 2.5. Two IVNSs 퐴 and 퐵 are equal if and Let 퐴 ⊂ 퐵 ⊂ 퐶, the we have only if 퐴 ⊆ 퐵 and 퐵 ⊆ 퐴. inf 푇퐴(푥푖) ≤ inf 푇퐵(푥푖) ≤ inf 푇퐶(푥푖) Definition 2.6. The union of two IVNSs 퐴 and 퐵 is an sup 푇퐴(푥푖) ≤ sup 푇퐵(푥푖) ≤ sup 푇퐶(푥푖) IVNS 퐶, written as 퐶 = 퐴 ∪ 퐵, is defined as follow inf 퐼퐴(푥푖) ≥ inf 퐼퐵(푥푖) ≥ inf 퐼퐶(푥푖) inf 푇퐶(푥) = max{inf 푇퐴(푥) , inf 푇퐵(푥)} sup 퐼퐴(푥푖) ≥ sup 퐼퐵(푥푖) ≥ sup 퐼퐶(푥푖) sup 푇퐶(푥) = max{sup 푇퐴(푥), sup 푇퐵(푥)} inf 퐹퐴(푥푖) ≥ inf 퐹퐵(푥푖) ≥ inf 퐹퐶(푥푖) inf 퐼퐶(푥) = min{inf 퐼퐴(푥) , inf 퐼퐵(푥)} sup 퐹퐴(푥푖) ≥ sup 퐹퐵(푥푖) ≥ sup 퐹퐶(푥푖) sup 퐼퐶(푥) = min{sup 퐼퐴(푥), sup 퐼퐵(푥)} for all 푥 ∈ 푋. It follows that inf 퐹퐶(푥) = min{inf 퐹퐴(푥) , inf 퐹퐵(푥)} |inf 푇퐴(푥푖) − inf 푇퐵(푥푖)| ≤ |inf 푇퐴(푥푖) − inf 푇퐶(푥푖)| sup 퐹퐶(푥) = min{sup 퐹퐴(푥), sup 퐹퐵(푥)} |sup 푇퐴(푥푖) − sup 푇퐵(푥푖)| ≤ |sup 푇퐴(푥푖) − sup 푇퐶(푥푖)| for all 푥 ∈ 푋. |inf 퐼퐴(푥푖) − inf 퐼퐵(푥푖)| ≤ |inf 퐼퐴(푥푖) − inf 퐼퐶(푥푖)| Definition 2.7. The intersection of two IVNSs 퐴 and 퐵, |sup 퐼퐴(푥푖) − sup 퐼퐵(푥푖)| ≤ |sup 퐼퐴(푥푖) − sup 퐼퐶(푥푖)| written as 퐶 = 퐴 ∩ 퐵, is defined as follow |inf 퐹퐴(푥푖) − inf 퐹퐵(푥푖)| ≤ |inf 퐹퐴(푥푖) − inf 퐹퐶(푥푖)| inf 푇퐶(푥) = min{inf 푇퐴(푥) , inf 푇퐵(푥)} |sup 퐹퐴(푥푖) − sup 퐹퐵(푥푖)| ≤ |sup 퐹퐴(푥푖) − sup 퐹퐶(푥푖)|. sup 푇퐶(푥) = min{sup 푇퐴(푥), sup 푇퐵(푥)} It means that 푆(퐴, 퐶) ≤ 푆(퐴, 퐵). Similarly, it seems inf 퐼퐶(푥) = max{inf 퐼퐴(푥) , inf 퐼퐵(푥)} that 푆(퐴, 퐶) ≤ 푆(퐵, 퐶). sup 퐼퐶(푥) = max{sup 퐼퐴(푥), sup 퐼퐵(푥)} The proof is completed inf 퐹퐶(푥) = max{inf 퐹퐴(푥) , inf 퐹퐵(푥)} sup 퐹퐶(푥) = max{sup 퐹퐴(푥), sup 퐹퐵(푥)} 4 Entropy of an interval valued neutrosophic set for all 푥 ∈ 푋. The entropy measure on IVIF sets is given by Wei [10]. 3 Similarity measure between interval valued neu- We extend the entropy measure on IVIF set to interval val- trosophic sets ued neutrosophic set. Definition 4.1. Let 푁(푋) be all IVNSs on 푋 and Similarity measure between Interval Neutrosophic sets is 퐴 ∈ 푁(푋). An entropy on IVNSs is a function 퐸푁: 푁(푋) → defined by Ye [11] as follow. [0,1] which is satisfies the following axioms: Definition 3.1. Let 푁(푋) be all IVNSs on 푋 and i. 퐸푁(퐴) = 0 if 퐴 is crisp set 퐴, 퐵 ∈ 푁(푋). A similarity measure between two IVNSs is ii. 퐸푁(퐴) = 1 if [inf 푇퐴(푥) , sup 푇퐴(푥)] = a function 푆: 푁(푋) ×푁(푋) → [0,1] which is satisfies the [inf 퐹퐴(푥) , sup 퐹퐴(푥)] and inf 퐼퐴(푥) = sup 퐼퐴(푥) following conditions: for all 푥 ∈ 푋 푐 i. 0 ≤ 푆(퐴, 퐵) ≤ 1 iii. 퐸푁(퐴) = 퐸푁(퐴 ) for all 퐴 ∈ 푁(푋) ii. 푆(퐴, 퐵) = 1 iff 퐴 = 퐵 iv. 퐸푁(퐴) ≥ 퐸푁(퐵) if 퐴 ⊆ 퐵 when sup 퐼퐴(푥) − iii. 푆(퐴, 퐵) = 푆(퐵, 퐴) sup 퐼퐵(푥) < inf 퐼퐴(푥) − inf 퐼퐵(푥) for all 푥 ∈ 푋. iv. If 퐴 ⊂ 퐵 ⊂ 퐶 then 푆(퐴, 퐶) ≤ 푆(퐴, 퐵) and Definition 4.2. The entropy of IVNS set 퐴 is, 푛 푆(퐴, 퐶) ≤ 푆(퐵, 퐶) for all 퐴, 퐵, 퐶 ∈ 푁(푋). 1 2 − |푖푛푓 푇퐴(푥) − 푖푛푓 퐹퐴(푥)| 퐸(퐴) = ∑ [ We generalize the similarity measure of intuitionistic 푛 푖=1 2 + |푖푛푓 푇퐴(푥) − 푖푛푓 퐹퐴(푥)| fuzzy set, which defined in [4], to interval valued neutro- −|푠푢푝 푇퐴(푥) − 푠푢푝 퐹퐴(푥)| − |푖푛푓 퐼퐴(푥) − 푠푢푝 퐼퐴(푥)| sophic sets as follow. ] +|푠푢푝 푇퐴(푥) − 푠푢푝 퐹퐴(푥)| + |푖푛푓 퐼퐴(푥) − 푠푢푝 퐼퐴(푥)| Definition 3.2. Let 퐴, 퐵 be two IVN sets in 푋. The for all 푥 ∈ 푋. similarity measure between the IVN sets 퐴 and 퐵 can be Theorem: The IVN entropy of 퐸 (퐴) is an entropy meas- evaluated by the function, 푁 ure for IVN sets. 푆(퐴, 퐵) = 1 − 1/2 [max(|inf 푇퐴(푥푖) − inf 푇퐵(푥푖)|) Proof: We show that the 퐸푁(퐴) satisfies the all proper- i + max(|sup 푇퐴(푥푖) − sup 푇퐵(푥푖)|) ties given in Definition 4.1. i + max(|inf 퐼퐴(푥푖) − inf 퐼퐵(푥푖)|) i. When 퐴 is a crisp set, i.e., i + max(|sup 퐼퐴(푥푖) − sup 퐼퐵(푥푖)|) [inf 푇퐴(푥) , sup 푇퐴(푥)] = [0,0] i + max(|inf 퐹퐴(푥푖) − inf 퐹퐵(푥푖)|) [inf 퐼퐴(푥) , sup 퐼퐴(푥)] = [0,0] i + max(|sup 퐹퐴(푥푖) − sup 퐹퐵(푥푖)|)] [inf 퐹퐴(푥) , sup 퐹퐴(푥)] = [1,1] i or [inf 푇퐴(푥) , sup 푇퐴(푥)] = [1,1] [inf 퐼퐴(푥) , sup 퐼퐴(푥)] = [0,0] for all 푥 ∈ 푋. [inf 퐹퐴(푥) , sup 퐹퐴(푥)] = [0,0] for all 푥푖 ∈ 푋. It is clear that 퐸푁(퐴) = 0.

Ali Aydoğdu, On Entropy and Similarity Measure of Interval Valued Neutrosophic Sets Neutrosophic Sets and Systems, Vol. 9, 2015 49

ii. Let [9] H. Wang, F. Smarandache, Y. Zhang, R. Sunderraman. Sin- th [inf 푇퐴(푥) , sup 푇퐴(푥)] = [inf 퐹퐴(푥) , sup 퐹퐴(푥)] gle Valued Neutrosophic Sets, in Proc. of 10 Int. Conf. on and inf 퐼퐴(푥) = sup 퐼퐴(푥) for all 푥 ∈ 푋. Then Fuzzy Theory and Technology, 2005. 푛 푛 [10] C-P. Wei, P. Wang, and Y-Z. Zhang. Entropy, similarity 1 2 − 0 − 0 − 0 1 measure of interval valued intuitionistic sets and their appli- 퐸 (퐴) = ∑ = ∑ 1 = 1. 푁 푛 2 + 0 + 0 + 0 푛 cations, Information Sciences, 181 (2011), 4273-4286. 푖=1 푖=1 [11] J. Ye. Similarity measures between interval neutrosophic iii. Since 푇퐴푐(푥) = 퐹퐴(푥), inf 퐼퐴푐(푥) = 1 − sup 퐼퐴(푥), sets and their applications in multicriteira decision-making, sup 퐼퐴푐(푥) = 1 − inf 퐼퐴(푥) and 퐹퐴푐(푥) = 푇퐴(푥) , Journal of Intelligent and Fuzzy Systems, 26 (2014), 2459- 푐 it is clear that 퐸푁(퐴) = 퐸푁(퐴 ). 2466. iv. If A⊆ 퐵, then inf 푇퐴(푥) ≤ inf 푇퐵(푥), sup 푇퐴(푥) ≤ [12] L. Zadeh. Fuzzy sets, Inform and Control, 8 (1965), 338- sup 푇퐵(푥) , inf 퐼퐴(푥) ≥ inf 퐼퐵(푥) , sup 퐼퐴(푥) ≥ 353. sup 퐼퐵(푥), inf 퐹퐴(푥) ≥ inf 퐹퐵(푥) and sup 퐹퐴(푥) ≥ sup 퐹 (푥). So 퐵 Received: June 2, 2015. Accepted: July 22, 2015. |inf 푇퐴(푥) − inf 퐹퐴(푥)| ≤ |inf 푇퐵(푥) − inf 퐹퐵(푥)| and |sup 푇퐴(푥) − sup 퐹퐴(푥)| ≤ |sup 푇퐵(푥) − sup 퐹퐵(푥)| for all 푥 ∈ 푋. And |inf 퐼퐴(푥) − sup 퐼퐴(푥)| ≤ |inf 퐼퐵(푥) − sup 퐼퐵(푥)| when sup 퐼퐴(푥) − sup 퐼퐵(푥) ≤ inf 퐼퐴(푥) − inf 퐼퐵(푥) for all 푥 ∈ 푋. Therefore it is clear that 퐸(퐴) ≥ 퐸(퐵). The proof is completed

Conclusion In this paper we introduced similarity measure of interval valued neutrosophic sets. These measures are con- sistent with similar considerations for other sets. Then we give entropy of an interval valued neutrosophic set. This entropy was generalized the entropy measure on interval valued intuitionistic sets in [10].

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Ali Aydoğdu, On Entropy and Similarity Measure of Interval Valued Neutrosophic Sets