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Book Series, Vol. 14, 2016 ISBN 978-1-59973-514-6 ISBN 978-1-59973-514-6 Neutrosophic Sets and Systems An International Book Series in Information Science and Engineering Quarterly Editor-in-Chief: Associate Editors: W. B. Vasantha Kandasamy, Indian Institute of Technology, Chennai, Tamil Nadu, India. Prof. Florentin Smarandache Said Broumi, Univ. of Hassan II Mohammedia, Casablanca, Morocco. Address: A. A. Salama, Faculty of Science, Port Said University, Egypt. Yanhui Guo, School of Science, St. Thomas University, Miami, USA. “Neutrosophic Sets and Systems” Francisco Gallego Lupiaňez, Universidad Complutense, Madrid, Spain. (An International Book Series in Peide Liu, Shandong University of Finance and Economics, China. Information Science and Engineering) Pabitra Kumar Maji, Math Department, K. N. University, WB, India. S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia. Department of Mathematics and Science Jun Ye, Shaoxing University, China. University of New Mexico Ştefan Vlăduţescu, University of Craiova, Romania. 705 Gurley Avenue Valeri Kroumov, Okayama University of Science, Japan. Gallup, NM 87301, USA Dmitri Rabounski and Larissa Borissova, independent researchers. Surapati Pramanik, Nandalal Ghosh B.T. College, West Bengal, India. E-mail: [email protected] Irfan Deli, Kilis 7 Aralık University, 79000 Kilis, Turkey. Home page: http://fs.gallup.unm.edu/NSS Rıdvan Şahin, Faculty of Science, Ataturk University, Turkey. Luige Vladareanu, Romanian Academy, Bucharest, Romania. Associate Editor-in-Chief: Mohamed Abdel-Baset,Faculty of computers and informatics, Zagazig University, Egypt. Mumtaz Ali A. A. A. Agboola, Federal University of Agriculture, Abeokuta, Nigeria. Address: Le Hoang Son, VNU Univ. of Science, Vietnam National Univ. Hanoi, Vietnam. Luu Quoc Dat, Univ. of Economics and Business, Vietnam National Univ., Hanoi, Vietnam. University of Southern Queensland 4300, Huda E. Khalid, University of Telafer, Telafer - Mosul, Iraq. Australia. Maikel Leyva-Vázquez, Universidad de Guayaquil, Guayaquil, Ecuador. Muhammad Akram, University of the Punjab, Lahore, Pakistan. Volume 14 2016 Contents Dragisa Stanujkic, Florentin Smarandache, Edmundas Rakib Iqbal, Sohail Zafar, Muhammad Shoaib Sardar. Kazimieras Zavadskas, Darjan Karabasevic. Multiple 3 Neutrosophic Cubic Subalgebras and Neutrosophic Cu- 47 Criteria Evaluation Model Based on the Single Valued bic Closed Ideals of B-algebras …….....………………. Neutrosophic Set …………………………………......... Pablo José Menéndez Vera, Cristhian Fabián Menéndez Huda E. Khalid, Florentin Smarandache, Ahmed K. Es- Delgado, Susana Paola Carrillo Vera, Milton Villegas 61 sa. A Neutrosophic Binomial Factorial Theorem with 7 Alava, Miriam Peña Gónzales. Static analysis in neu- their Refrains ………………………………….............. trosophic cognitive maps ………………...........………. Kul Hur, Pyung Ki Lim, Jeong Gon Lee, Junhui Kim. Nguyen X. Thao, Florentin Smarandache. (I,T)- The Category of Neutrosophic Sets ...…………………. 12 Standard neutrosophic rough set and its topologies 65 properties ………..................………………………….. Harish Garg, Nancy. On Single-Valued Neutrosophic Entropy of order α ......…………………………………. 21 Naga Raju I, Rajeswara Reddy P, Dr. Diwakar Reddy V, Dr. Krishnaiah G. Real Life Decision Optimization 71 Salah Bouzina. Fuzzy Logic vs. Neutrosophic Logic: Model .......………..................…………………………. Operations Logic ……………………………….......…. 29 Nguyen Xuan Thao, Bui Cong Cuong, Florentin Rajashi Chatterjee, Pinaki Majumdar, Syamal Kumar Smarandache. Rough Standard Neutrosophic Sets: An 80 Samanta. Interval-valued Possibility Quadripartitioned 35 Application on Standard Neutrosophic Information Single Valued Neutrosophic Soft Sets and some uncer- Systems …………………………………....................... tainty based measures on them ………………………... Wenzhong Jiang, Jun Ye: Optimal Design of Truss W.B. Vasantha Kandasamy, K. Ilanthenral, Florentin Structures Using a Neutrosophic Number Optimization 93 Smarandache. Modified Collatz conjecture or (3a + 1) 44 Model under an Indeterminate Environment .........……. + (3b + 1)I Conjecture for Neutrosophic Numbers ...…. The Educational Publisher Inc. 1313 Chesapeake Ave. Columbus, Ohio 43212, USA. Copyright © Neutrosophic Sets and Systems Neutrosophic Sets and Systems, Vol. 14, 2016 Neutrosophic Sets and Systems An International Book Series in Information Science and Engineering Copyright Notice Copyright @ Neutrosophics Sets and Systems ademic or individual use can be made by any user without All rights reserved. The authors of the articles do hereby permission or charge. 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Copyright © Neutrosophic Sets and Systems Neutrosophic Sets and Systems, Vol. 14, 2016 3 University of New Mexico Multiple Criteria Evaluation Model Based on the Single Valued Neutrosophic Set Dragisa Stanujkic1, Florentin Smarandache2, Edmundas Kazimieras Zavadskas3 and Darjan Karabasevic4 1Faculty of Management in Zajecar, John Naisbitt University, Goce Delceva 8, Belgrade 11070, Serbia. E-mail: [email protected] 2Department of Mathematics, University of New Mexico, 705 Gurley Avenue, Gallup, NM 87301, USA. E-mail: [email protected] 3Research Institute of Smart Building Technologies, Civil Engineering Faculty Vilnius Gediminas Technical University, Saulėtekio al. 11, Vilnius 10221, Lithuania. E-mail: [email protected] 4Faculty of Applied Management, Economics and Finance, University Business Academy in Novi Sad, Jevrejska 24, 11000, Belgrade, Serbia. E-mail: [email protected] Abstract. Gathering the attitudes of the examined re- An example of the evaluation of restaurants is considered spondents would be very significant in some evaluation at the end of this paper with the aim to present in detail models. Therefore, a multiple criteria approach based on the proposed approach. the use of the neutrosophic set is considered in this paper. Keywords: neutrosophic set, single valued neutrosophic set, multiple criteria evaluation. 1.Introduction tion is based on the ratings generated from respondents, the In order to deal with indeterminate and incon- NS and the SVNS can provide some advantages in relation sistent information, Smarandache [1] proposed a to the usage of crisp and other forms of fuzzy numbers. neutrosophic set (NS), thus simultaneously providing Therefore, the rest of this paper is organized as fol- a general framework generalizing the concepts of the clas- lows: in Section 2, some basic definitions related to the sical, fuzzy [2], interval-valued [3, 4], intuitionistic [5] SVNS are given. In Section 3, an approach to the deter- and interval-valued intuitionistic [6] fuzzy sets. mining of criteria weights is presented, while Section 4 The NS has been applied in different fields, such as: proposes a multiple criteria evaluation model based on the the database [7], image processing [8, 9, 10], the medical use of the SVNS. In Section 5, an example is considered diagnosis [11, 12], decision making [13, 14], with a partic- with the aim to explain in detail the proposed methodology. ular emphasis on multiple criteria decision making [15, 16, The conclusions are presented at the end of the manuscript. 17, 18, 19, 20]. In addition to the membership function, or the so- 2.The Single Valued Neutrosophic Set called truth-membership T (x), proposed in fuzzy sets, At- A anassov [5] introduced the non-membership function, or Definition 1. [21] Let X be the universe of discourse, the so-called falsity-membership F (x), which expresses with a generic element in X denoted by x. Then, the Neu- A non-membership to a set, thus creating the basis for the trosophic Set (NS) A in X is as follows: solving of a much larger number of decision-making prob- A{xT (x),I (x),F (x)|xX}, (1) lems. A A A In intuitionistic fuzzy sets, the indeterminacy I (x)is A where T (x), I (x) and F (x) are the truth-membership 1T (x)F (x) by default. A A A A A function, the indeterminacy-membership function and the In the NS, Smarandache [21] introduced independent falsity-membership function, respectively, indeterminacy-membership IA(x) , thus making the NS T ,I ,F :X ]0,1[ and 0 T(x)+I(x)+U(x) more flexible and the most suitable for solving some com- A3A A A A A plex decision-making problems, especially decision- Definition 2. [1, 22] Let X be the universe of dis- making problems related to the use of incomplete and im- course. The Single Valued Neutrosophic Set (SVNS) A precise information, uncertainties and predictions and so over X is an object having the form: on. Smarandache [1] and Wang et al. [22] further pro- A{xT (x),I (x),F (x)|xX}, (2) A A A posed the single valued neutrosophic set (SVNS) suitable for solving many real-world decision-making problems. where T (x), I (x) and F (x) are the truth-membership A A A In multiple criteria evaluation models, where evalua- function, the intermediacy-membership function and the Dragisa Stanujkic, Florentin Smarandache, Edmundas Kazimieras Zavadskas and Darjan Karabasevic, Multiple Criteria Evaluation Model Based on the Single Valued Neutrosophic Set 4 Neutrosophic Sets and Systems, Vol. 14, 2016 falsity-membership function, respectively, 1 when significance of C C TA,IA,FA:X [0,1]and 0 ≤ TA(x)+IA(x)+UA(x) ≤ 3. s  1 when significanse of Cj Cj1. (9) Definition 3. [21] For an SVNS A in X, the tri- j j j1  pletA,iA, fA is called the single valued neutrosophic 1 when significance of Cj Cj1 number (SVNN). By using Eq. (9), respondents are capable of express- Definition 4. SVNNs. Let x t , i ,f  and 1 1 1 1 ing their opinions more realistically compared to the ordi- x t , i ,f  be two SVNNs and0; then, the basic 2 2 2 2 nary SWARA method, proposed by Kersuliene et al. [25]. operations are defined as follows: Step 3. The third step in the adapted SWARA method x1x2 t1t2t1t2,i1i2,f1f2 . (3) should be performed as follows: x x tt ,i i ii f  f  f f . (4)  1 j1 1 2 12 1 2 12, 1 2 1 2 k  . (10) j 2s j1 x 1(1t ),i, f. (5)  j 1 1 1 1 where k is a coefficient. j xt,i,1(1 f ). (6) 1 1 1 1 Step 4. Determine the recalculated weight q as fol- j lows: Definition 5. [23] Let xt , i ,f  be a SVNN; x x x then the cosine similarity measure S between SVNN x (x)  1 j1 and the ideal alternative (point) <1,0,0> can be defined as q  . (11) j q k j1 follows:  j1 j t Step 5. Determine the relative weights of the evalua- Sx . (7) tion criteria as follows: t2i2 f2 w q n q , (12) Definition 6. [23] Let A t , i , f  be a collection j j k1 k j j j j of SVNSs and W (w,w ,...,w )T be an associated 1 2 n where wj denotes the relative weight of the criterion j. weighting vector. Then the Single Valued Neutrosophic Weighted Average (SVNWA) operator of A is as follows: j 4. A Multiple Criteria Evaluation Model Based on n the Use of the SVNS SVNWA(A,A ,...,A ) w A 1 2 n j j j1 For a multiple criteria evaluation problem involving , (8) the m alternatives that should be evaluated by the K re- 1n (1t )wj,n (i )wj,n (f )wj spondents based on the n criteria, whereby the performanc-  j j j   j1 j1 j1  es of alternatives are expressed by using the SVNS, the calculation procedure can be expressed as follows: where: w is the element j of the weighting vector, j The determination of the criteria weights. The deter- w [0, 1] andn w 1. j j1 j mination of the criteria weights can be done by applying various methods, for example by using the AHP method. 3.The SWARA Method However, in this approach, it is recommended that the The Step-wise Weight Assessment Ratio Analysis SWARA method should be used due to its simplicity and a (SWARA) technique was proposed by Kersuliene et al. smaller number of pairwise comparisons compared with [25]. The computational procedure of the adapted SWARA the well-known AHP method. method can be shown through the following steps: The determination of the criteria weight is done by using an interactive questionnaire made in a spreadsheet file. Step 1. Determine the set of the relevant evaluation By using such an approach, the interviewee can see the criteria and sort them in descending order, based on their calculated weights of the criteria, which enables him/her expected significances. modify his or her answers if he or she is not satisfied with Step 2. Starting from the second criterion, determine the calculated weights. the relative importance s of the criterion j in relation to the j Gathering the ratings of the alternatives in relation to previous (j-1) criterion, and do so for each particular crite- the selected set of the evaluation criteria. Gathering the rion as follows: ratings of the alternatives in relation to the chosen set of criteria is also done by using an interactive questionnaire. In this questionnaire, a declarative sentence is formed for each one of the criteria, thus giving an opportunity to the Dragisa Stanujkic, Florentin Smarandache, Edmundas Kazimieras Zavadskas and Darjan Karabasevic, Multiple Criteria Evaluation Model Based on the Single Valued Neutrosophic Set Neutrosophic Sets and Systems, Vol. 14, 2016 5 respondents to fill in their attitudes about the degree of The attitudes obtained from the three surveys, as well truth, indeterminacy and falsehood of the statement. as the appropriate weights, are accounted for in Table 2. The formation of the separated ranking order based on the weights and ratings obtained from each respond- E1 E1 E1 ent. At this steep, the ranking order is formed for each one sj wj sj wj sj wj of the respondents, based on the respondent’s respective C1 0.15 0.16 0.19 weights and ratings, in the following manner: C2 1.00 0.15 1.00 0.16 1.00 0.19  the determination of the overall ratings expressed C3 1.15 0.18 1.20 0.20 1.05 0.20 in the form of the SVNN by using Eq. (8), for C4 1.30 0.26 1.10 0.22 1.10 0.22 each respondent; C5 1.00 0.26 1.10 0.25 0.95 0.21  the determination of the cosine similarity measure, Table 2. The attitudes and the weights obtained from the three surveys for each respondent; and  the determination of the ranking order, for each The ratings of the alternatives expressed in terms of the respondent. SVNS obtained on the basis of the three surveys are given The determination of the most appropriate alternative. in Tables 3 to 5. Contrary to the commonly used approach in group decision making, no group weights and ratings are used in this ap- C1 C2 C3 C4 C5 proach. As a result of that, there are the K ranking orders w 0.15 0.15 0.18 0.26 0.26 of the alternatives and the most appropriate alternative is j A <0.8,0.1,0.3> <0.7,0.2,0.2> <0.8,0.1,0.1> <1,0.01,0.01> <0.8,0.1,0.1> the one determined on the basis of the theory of dominance 1 [26]. A2 <0.7,0.1,0.2> <1.0,0.1,0.1> <1.0,0.2,0.1> <1,0.01,0.01> <0.8,0.1,0.1> A <0.7,0.1,0.1> <1.0,0.1,0.1> <0.7,0.1,0.1> <0.9,0.2,0.01> <0.9,0.1,0.1> 3 5.A Numerical Illustration A <0.7,0.3,0.3> <0.7,0.1,0.1> <0.8,0.1,0.2> <0.9,0.1,0.1> <0.9,0.1,0.1> 4 In this numerical illustration, some results adopted Table 3. The ratings obtained based on the first survey from a case study are used. In the said study, four traditional restaurants were evaluated based on the following C1 C2 C3 C4 C5 criteria: w 0.16 0.16 0.20 0.22 0.25 j  the interior of the building and the friendly at- A <0.8,0.1,0.4> <0.9,0.15,0.3> <0.9,0.2,0.2> <0.85,0.1,0.25> <1.0,0.1,0.2> 1 mosphere, A <0.9,0.15,0.3> <0.9,0.15,0.2> <1.0,0.3,0.2> <0.7,0.2,0.1> <0.8,0.2,0.3> 2  the helpfulness and friendliness of the staff, A <0.6,0.15,0.3> <0.55,0.2,0.3> <0.55,0.3,0.3> <0.6,0.3,0.2> <0.7,0.2,0.3>  the variety of traditional food and drinks, 3 A <0.6,0.4,0.5> <0.6,0.3,0.1> <0.6,0.1,0.2> <0.7,0.1,0.3> <0.5,0.2,0.4>  the quality and the taste of the food and drinks, 4 Table 4. The ratings obtained based on the second survey including the manner of serving them, and  the appropriate price for the quality of the services provided. C1 C2 C3 C4 C5 The survey was conducted via e-mail, using an interac- wj 0.19 0.19 0.20 0.22 0.21 tive questionnaire, created in a spreadsheet file. By using A1 <1.0,0.1,0.1> <0.9,0.15,0.2> <1.0,0.2,0.1> <0.8,0.1,0.1> <0.9,0.1,0.2> such an approach, the interviewee could see the calculated A <0.8,0.15,0.3> <0.9,0.15,0.2> <1,0.2,0.2> <0.7,0.2,0.1> <0.8,0.2,0.3> 2 weights of the criteria and was also able to modify his/her A <0.6,0.15,0.3> <0.55,0.2,0.3> <0.55,0.3,0.3> <0.6,0.3,0.2> <0.7,0.2,0.3> 3 answers if he or she was not satisfied with the calculated A <0.8,0.4,0.5> <0.6,0.3,0.1> <0.6,0.4,0.1> <0.7,0.1,0.3> <0.5,0.2,0.4> 4 weights. Table 5. The ratings obtained from the third of the third survey In order to explain the proposed approach, three com- pleted surveys have been selected. The attitudes related to The calculated overall ratings obtained on the basis of the weights of the criteria obtained in the first survey are the first of the three surveys expressed in the form of shown in Table 1. Table 1 also accounts for the weights of SVNSs are presented in Table 6. The cosine similarity the criteria. measures, calculated by using Eq. (7), as well as the ranking order of the alternatives, are accounted for in Table 6. Criteria sj kj qj wj C1 1 1 0.15 Overall ratings Si Rank C2 1.00 1.00 1.00 0.15 A1 <1.0,0.06,0.07> 0.995 2 C3 1.15 0.85 1.18 0.18 A2 <1.0,0.06,0.06> 0.996 1 C4 1.30 0.70 1.68 0.26 A3 <1.0,0.12,0.06> 0.991 3 C5 1.00 1.00 1.68 0.26 A4 <1.0,0.12,0.13> 0.978 4 Table 1. The attitudes and the weights of the criteria obtained on the basis Table 6. The ranking orders obtained on the basis of the ratings of the of the first of the three surveys first survey Dragisa Stanujkic, Florentin Smarandache, Edmundas Kazimieras Zavadskas and Darjan Karabasevic, Multiple Criteria Evaluation Model Based on the Single Valued Neutrosophic Set 6 Neutrosophic Sets and Systems, Vol. 14, 2016 The ranking orders obtained based on all the three surveys are accounted for in Table 7. E1 E2 E3 E1 E2 E3 [10] Y. Guo, and H. D. Cheng. New neutrosophic approach to image segmentation. Pattern Recognition, 42, (2009), 587– Si Si Si Rank Rank Rank 595. A1 0.995 0.963 0.985 2 1 1 [11] S. Broumi, and I. Deli. Correlation measure for neutrosoph- A2 0.996 0.962 0.966 1 2 2 ic refined sets and its application in medical diagnosis. Pal- A3 0.991 0.864 0.867 3 4 4 estine Journal of Mathematics, 3(1), (2014), 11-19. [12] Ansari, A. Q., Biswas, R., and Aggarwal, S. (2011). Pro- A4 0.978 0.882 0.894 4 3 3 posal for applicability of neutrosophic set theory in medical Table 7. The ranking orders obtained from the three examinees AI. International Journal of Computer Applications, 27(5), 5-11. According to Table 7, the most appropriate alternative [13] A. Kharal. A Neutrosophic Multicriteria Decision Making based on the theory of dominance is the alternative denoted Method, New Mathematics and Natural Computation, as A1. Creighton University, USA, 2013. [14] J. Ye. 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Accepted: November 4, 2016 Dragisa Stanujkic, Florentin Smarandache, Edmundas Kazimieras Zavadskas and Darjan Karabasevic, Multiple Criteria Evaluation Model Based on the Single Valued Neutrosophic Set Neutrosophic Sets and Systems, Vol. 14, 2016 7 University of New Mexico A Neutrosophic Binomial Factorial Theorem with their Refrains Huda E. Khalid1 Florentin Smarandache2 Ahmed K. Essa3 1 University of Telafer, Head of Math. Depart., College of Basic Education, Telafer, Mosul, Iraq. E-mail: [email protected] 2 University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA. E-mail: [email protected] 3 University of Telafer, Math. Depart., College of Basic Education, Telafer, Mosul, Iraq. E-mail: [email protected] Abstract. The Neutrosophic Precalculus and the Two other important theorems were proven with their Neutrosophic Calculus can be developed in many corollaries, and numerical examples as well. As a ways, depending on the types of indeterminacy one conjecture, we use ten (indeterminate) forms in has and on the method used to deal with such neutrosophic calculus taking an important role in indeterminacy. This article is innovative since the limits. To serve article's aim, some important form of neutrosophic binomial factorial theorem was questions had been answered. constructed in addition to its refrains. Keyword: Neutrosophic Calculus, Binomial Factorial Theorem, Neutrosophic Limits, Indeterminate forms in Neutrosophic Logic, Indeterminate forms in Classical Logic. 1 Introduction (Important questions) 2. Let Q 1 What are the types of indeterminacy? 3√𝐼 =𝑥+𝑦 𝐼 0+𝐼 =𝑥3+3𝑥2𝑦 𝐼+3𝑥𝑦2 𝐼2+𝑦3𝐼3 There exist two types of indeterminacy 0+𝐼 =𝑥3+(3𝑥2𝑦 +3𝑥𝑦2 +𝑦3)𝐼 a. Literal indeterminacy (I). As example: 𝑥 =0,𝑦=1→ 3√𝐼 =𝐼. (5) 2+3𝐼 (1) In general, b. Numerical indeterminacy. 2𝑘+1√𝐼 =𝐼, (6) As example: where 𝑘 ∈𝑧+ ={1,2,3,…}. 𝑥(0.6,0.3,0.4)∈𝐴, (2) Basic Notes meaning that the indeterminacy membership = 0.3. 1 . A component I to the zero power is Other examples for the indeterminacy com- undefined value, (i.e. 𝐼0 is undefined), ponent can be seen in functions: 𝑓(0)=7 𝑜𝑟 9 or since 𝐼0 =𝐼1+(−1) =𝐼1∗𝐼−1 =𝐼 which is 𝑓(0 𝑜𝑟 1)=5 or 𝑓(𝑥)=[0.2, 0.3] 𝑥2 … etc. 𝐼 impossible case (avoid to divide by 𝐼). 2. The value of 𝐼 to the negative power is Q 2 What is the values of 𝐼 to the rational power? undefined value (i.e. 𝐼−𝑛 ,𝑛>0 is 1. Let undefined). √𝐼 =𝑥+𝑦 𝐼 Q 3 What are the indeterminacy forms in neutros- 0+𝐼 =𝑥2+(2𝑥𝑦+𝑦2)𝐼 ophic calculus? 𝑥 =0,𝑦=±1. (3) In classical calculus, the indeterminate forms In general, are [4]: 2𝑘√𝐼 =±𝐼 (4) 0 ,∞,0∙∞ ,∞0,00,1∞,∞−∞. (7) 0 ∞ where 𝑘 ∈𝑧+ ={1,2,3,…}. Huda E. Khalid, Florentin Smarandache & Ahmed K. Essa, A Neutrosophic Binomial Factorial Theorem with their Refrains 8 Neutrosophic Sets and Systems, Vol. 14, 2016 The form 0 to the power 𝐼 (i.e. 0𝐼 ) is an [2.1,2.5] [2.1,2.5] ∴limln𝑦=lim = indeterminate form in Neutrosophic calculus; it is 𝑥→0 𝑥→0 1 1 tempting to argue that an indeterminate form of ln𝑥 ln0 [2.1,2.5] [2.1,2.5] type 0𝐼 has zero value since "zero to any power is = = 1 −0 zero". However, this is fallacious, since 0𝐼 is not a −∞ power of number, but rather a statement about 2.1 2.5 =[ , ]=(−∞,−∞) limits. −0 −0 =−∞ Q 4 What about the form 1𝐼? Hence 𝑦=𝑒−∞ =0 The base "one" pushes the form 1𝐼 to one OR it can be solved briefly by while the power 𝐼 pushes the form 1𝐼 to I, so 1𝐼 is 𝑦=𝑥[2.1,2.5] =[02.1,02.5]=[0,0]=0. an indeterminate form in neutrosophic calculus. Example (3.2) Indeed, the form 𝑎𝐼, 𝑎 ∈ 𝑅 is always an lim [3.5,5.9]𝑥[1,2] = [3.5,5.9] [9,11][1,2] = indeterminate form. 𝑥→[9,11] Q 5 What is the value of 𝑎𝐼 ,𝑤 ℎ𝑒𝑟𝑒 𝑎∈𝑅? [3.5,5.9] [91,112]= [(3.5)(9),(5.9)(121)]= Let 𝑦 =2𝑥 ,𝑥 ∈𝑅 ,𝑦 =2𝐼; it is obvious that [31.5,713.9]. 1 2 lim 2𝑥 =∞ , lim 2𝑥 =0 ,lim2𝑥 =1; while Example (3.3) 𝑥→∞ 𝑥→−∞ 𝑥→0 we cannot determine if 2𝐼 →∞ 𝑜𝑟 0 𝑜𝑟 1, lim[3.5,5.9] 𝑥[1,2] =[3.5,5.9] ∞[1,2] 𝑥→∞ therefore we can say that 𝑦2 =2𝐼 indeterminate =[3.5,5.9] [∞1,∞2] form in Neutrosophic calculus. The same for 𝑎𝐼, = [3.5∙(∞) ,5.9∙(∞)] where 𝑎∈𝑅 [2]. =(∞,∞)=∞. 2 Indeterminate forms in Neutrosophic Example (3.4) Logic Find the following limit using more than one It is obvious that there are seven types technique lim√[4,5]∙𝑥+1−1 . of indeterminate forms in classical calculus [4], 𝑥→0 𝑥 Solution: 0,∞,0.∞,00,∞0,1∞,∞−∞. The above limit will be solved firstly by using the 0 ∞ L'Hôpital's rule and secondly by using the As a conjecture, we can say that there are ten rationalizing technique. forms of the indeterminate forms in Neutrosophic Using L'Hôpital's rule calculus 1 𝐼 ∞ lim ([4,5]∙𝑥+1)−1⁄2 [4,5] 𝐼0 ,0𝐼, ,𝐼∙∞, ,∞𝐼,𝐼∞,𝐼𝐼, 𝑥→02 0 𝐼 [4,5] 𝑎𝐼(𝑎∈𝑅),∞±𝑎∙𝐼 . =lim 𝑥→02√([4,5]∙𝑥+1) Note that: [4,5] 4 5 = = [ , ]= [2,2.5] 2 2 2 𝐼 1 =𝐼∙ =𝐼∙∞=∞∙𝐼. 0 0 Rationalizing technique [3] √[4,5]∙𝑥+1−1 √[4,5]∙0+1−1 3 Various Examples lim = Numerical examples on neutrosophic limits 𝑥→0 𝑥 0 would be necessary to demonstrate the aims of this √[4∙0,5∙0]+1−1 √[0,0]+1−1 = = work. 0 0 √0+1−1 0 Example (3.1) [1], [3] = = 0 0 The neutrosophic (numerical indeterminate) values =undefined. can be seen in the following function: Find lim𝑓(𝑥), where 𝑓(𝑥)=𝑥[2.1,2.5]. Multiply with the conjugate of the numerator: 𝑥→0 Solution: Let 𝑦=𝑥[2.1,2.5] →ln𝑦=[2.1,2.5] ln𝑥 Huda E. Khalid, Florentin Smarandache & Ahmed K. Essa, A Neutrosophic Binomial Factorial Theorem with their Refrains Neutrosophic Sets and Systems, Vol. 14, 2016 9 √[4,5]𝑥+1−1 √[4,5]𝑥+1+1 Again, Solving by using L'Hôpital's rule lim ∙ 𝑥→0 𝑥 √[4,5]𝑥+1+1 𝑥2+3𝑥−[1,2]𝑥−[3,6] lim (√[4,5]𝑥+1)2−(1)2 𝑥→−3 𝑥+3 2 𝑥+3−[1,2] =lim = lim 𝑥→0 𝑥(√[4,5]𝑥+1+1) 𝑥→−3 1 2 (−3)+3−[1,2] [4,5]∙𝑥+1−1 = lim =lim 𝑥→−3 1 𝑥→0𝑥∙(√[4,5]𝑥+1+1) =−6+3−[1,2] [4,5]∙𝑥 =−3−[1,2] =lim 𝑥→0𝑥∙(√[4,5]𝑥+1+1) =[−3−1,−3−2] =[−5,−4] [4,5] =lim The above two methods are identical in results. 𝑥→0(√[4,5]𝑥+1+1) [4,5] [4,5] 4 New Theorems in Neutrosophic Limits = = (√[4,5]∙0+1+1) √1+1 Theorem (4.1) (Binomial Factorial ) =[4,5]=[4,5]=[2,2.5]. lim(𝐼+1)𝑥=𝐼𝑒 ; I is the literal indeterminacy, 2 2 2 𝑥→∞ 𝑥 e = 2.718282 8 Identical results. Proof Example (3.5) 1 𝑥 𝑥 1 0 𝑥 1 1 Find the value of the following neutrosophic limit (𝐼+ ) =( )𝐼𝑋( ) +( )𝐼𝑋−1( ) 𝑥 0 𝑥 1 𝑥 𝑥2+3𝑥−[1,2]𝑥−[3,6] lim using more than one 𝑥 1 2 𝑥 1 3 𝑥→−3 𝑥+3 +( )𝐼𝑋−2( ) +( )𝐼𝑋−3( ) technique . 2 𝑥 3 𝑥 𝑥 1 4 Analytical technique [1], [3] +( )𝐼𝑋−4( ) +⋯ 4 𝑥 𝑥2+3𝑥−[1,2]𝑥−[3,6] lim 1 𝐼 1 𝑥→−3 𝑥+3 =𝐼+𝑥.𝐼. + (1− ) By substituting 𝑥= -3 , 𝑥 2! 𝑥 𝐼 1 2 𝐼 1 2 (−3)2+3∙(−3)−[1,2]∙(−3)−[3,6] + (1− )(1− )+ (1− )(1− ) lim 3! 𝑥 𝑥 4! 𝑥 𝑥 𝑥→−3 −3+3 3 9−9−[1∙(−3),2∙(−3)]−[3,6] (1− )+⋯ = 𝑥 0 1 0−[−6,−3]−[3,6] [3,6]−[3,6] It is clear that →0 𝑎𝑠 𝑥 →∞ = = 𝑥 0 0 ∴lim(𝐼−1)𝑥 =𝐼+𝐼+ 𝐼 + 𝐼 + 𝐼 +⋯=𝐼+ [3−6,6−3] [−3,3] 𝑥→∞ 𝑥 2! 3! 4! = 0 = 0 , ∑∞𝑛=1𝐼𝑛𝑛! which has undefined operation00,since 0∈ ∴𝑥→li∞m(𝐼+𝑥1)𝑥=𝐼𝑒, where e =1+∑∞𝑛=1𝑛1! , I is the [−3,3]. Then we factor out the numerator, and literal indeterm inacy. simplify: 𝑥2+3𝑥−[1,2]𝑥−[3,6] Corollary (4.1.1) lim = 𝑥→−3 𝑥+3 1 (𝑥−[1,2])∙(𝑥+3) lim(𝐼+𝑥)𝑥 =𝐼𝑒 lim = lim(𝑥−[1,2]) 𝑥→0 𝑥→−3 (𝑥+3) 𝑥→−3 Proof:- =−3−[1,2]=[−3,−3]−[1,2] Put 𝑦=1 = −([3,3]+[1,2])=[−5,−4]. 𝑥 It is obvious that 𝑦→∞ , as 𝑥 →0 ∴lim(𝐼+𝑥)𝑥1 = lim(𝐼+1)𝑦 =𝐼𝑒 𝑥→0 𝑦→∞ 𝑦 ( using Th. 4.1 ) Corollary (4.1.2) lim(𝐼+𝑘)𝑥=𝐼𝑒𝑘 , where k >0 & 𝑘≠0 , I is the 𝑥→∞ 𝑥 literal indete rminacy. Huda E. Khalid, Florentin Smarandache & Ahmed K. Essa, A Neutrosophic Binomial Factorial Theorem with their Refrains