• Bui Cong Cuong Institute of Mathematics, Vietnam Academy of Science and Technology



Pythagorean picture fuzzy set, Picture logic operators, Decision-making problems


Pythagorean picture fuzzy set (PPFS) - is a combination of Picture fuzzy set with the Yager’s Pythagorean fuzzy set [12-14]. In the first part of  the paper [17] we considered basic notions on PPFS as set operators of PPFS. Unfortunately, we have not papers [18,19, 20]  about spherical fuzzy sets with the same definition with some operators and applications to multi attribute group decision making problems. Now in the second part, we will present some main operators in picture fuzzy logic on PPFS: picture negation operator, picture t-norm, picture t-conorm, picture implication operators on PPFS. Last, the compositional rule of inference in PPFS setting should be presented and an  numerical example was given.


[1] L.A. Zadeh, Fuzzy Sets, Information and Control, 8, (1965) 338-353.

[2] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Information Sciences, vol. 8, (1975) 199-249.

[3] K.Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, vol.20 (1986) 87-96.

[4] K.Atanassov, On Intuitionistic Fuzzy Sets Theory, Springer, 2012, Berlin.

[5] T.H.Nguyen and E.Walker, A first course in fuzzy logic, Second Edition, Chapman& Hall CRC, 2000, Boca Raton,

[6] E.P.Klement and R.Mesiar, Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms, 1st Edition, 2005.

[7] J.Fodor and M. Roubens, Fuzzy preference modeling and multicriteria decision support, Kluwer Academic Pub. London, 1994.

[8] B.C.Cuong and V.Krenovitch, Picture Fuzzy Sets - a new concept for computational intelligence problems, in the Proceedings of the 3rd World Congress on Information and Communication Technology, WICT 2013, Hanoi, IEEE CS, pp 1-6, 2013.

[9] B.C. Cuong, Picture fuzzy sets, Journal Computer science and Cybernetics, V. 30(4), 419-430, 2014.

[10] B.C.Cuong, V.Kreinovich, R.T.Ngan, A classification of representable t-norms operators for

picture fuzzy sets, 2016, Departmental Technical Reports (CS), Paper 1047,

[11] B.C.Cuong, R.T.Ngan, L.B. Long, Some new de morgan picture operator triples in picture fuzzy logic, Journal of computer science and Cybernetics, V. 33(2),143-164,2017.

[12] A.A.Yager, Pythagorean fuzzy subsets, in the Proceeding of Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, Canada, 57-61, 2013.

[13] A.A.Yager, A.M. Abbasov, Pythagorean membership grades, complex numbers, and decision making, Int. J. Intell. Syst., 28(5), 436-452, 2013.

[14] A.A. Yager, Pythagorean membership grades in mylticriteria decision making, IEEE Trans. Fuzzy Syst., 22(4), 958-965,2014.

[15] D. Molodtsov, Soft set theory – First results, Computer & Mathematics with Applications 37 (4-5), 19-31,1999

[16] P.K.Maji, R.Bismas, A.R.Roy, Soft set theory, Computer & Mathematics with Applications 45 555-562,2003

[17] B.C. Cuong, Pythagorean Picture Fuzzy Sets, Part 1 - Basic notions, Journal of computer science and Cybernetics, vol.35. n4, 2019,

[18] S.Ashraf, S, Abdulallsh, Spherical aggregation operator and their application in multiattribute. group decision making, Int. J. Intell. Syst. , 2018, 1-4, http://

[19] S. Áshraf, S. Abdullah, M. Aslam, M. Qiyas, M.Kutbi, Sphrical fuzzy sets and ít repreentation of sherical fuzzy t-norms and t-conorms, Journal of Intelligent &Fuzzy Systems ,36, 2019, 6089-6102.DOS. A I 10.3232, JIFS 181941

[20] S. Ashraf. S.ASbdullah, T. Mahmood, F. Ghan, Spherical fuzzy sets and their applications in multi -attribute decision making problems, . DOI 10.3233/JIFS 172009

[21] M.Wu, T. Chu, H. Fan, Divergence measures of T- spherical fuzzy sét and its application in pattern recognition, IEEE Access, vol. 8, 2020,10208-10221

[22] P.H. Phong, B.C. Cuong, Some intuitionistic linguistic aggregation operators, Journal of Computer Science and Cybernetics, vol. 30, n.3, 216-226, 2014.

[23] P.H.Phong, B.C.Cuong, Multi-criteria group decision making with Picture Linguistic Numbers, VNU Journal of Science: Computer Science and Communication Engineering, vol.32, n.3, 38-51, 2016.

[24] L.H.Son.,Thong, P.H.: Some Novel Hybrid Forecast Methods Based On Picture Fuzzy Clustering for Weather Nowcasting from Satellite Image Sequences. Applied Intelligence 46(1) (2017) 1-15.

[25] L.H.Son, Viet, P.V., Hai, P.V. Picture Inference System: A New Fuzzy Inference System on Picture Fuzzy Set. Applied Intelligence (2017) DOI: 10.1007/s10489-016-0856-1.

[26] L.H.Son, L DPFCM: A novel distributed picture fuzzy clustering method on picture fuzzy sets. Expert systems with applications 42 (2015) 51-66.

[27] P.H.Thong and L.H.Son, Picture Fuzzy Clustering: A New Computational Intelligence Method, Soft Computing, v.20 (9) 3544-3562, 2016.