Some New De Morgan Picture Operator Triples in Picture Fuzzy Logic

Cuong Bui Cong, Roan Thi Ngan, Le Ba Long


A new concept of picture fuzzy sets (PFS) were introduced in 2013, which are direct
extensions of the fuzzy sets and the intuitonistic fuzzy sets. Then some operations on PFS with
some properties are considered in [ 9,10 ]. Some basic operators of fuzzy logic as negation, tnorms
, t-conorms for picture fuzzy sets firstly are defined and studied in [13,14]. This paper is
devoted to some classes of representable picture fuzzy t-norms and representable picture fuzzy
t-conorms on PFS and a basic algebra structure of Picture Fuzzy Logic – De Morgan triples of
picture operators.


Key words: Picture fuzzy sets,, Picture fuzzy t-norms, Picture fuzzy t-conorm , De Morgan picture operator triple

Full Text:



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

L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning ,

Information Sciences, vol. 8, (1975) 199-249.

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

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

T.H.Nguyen and E.Walker, A first course in fuzzy logic , Second Edition, Chapman& Hall /

CRC, 2000, Boca Raton,

E.P.Klement and R.Mesiar , Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular

Norms, 1st Edition, 2005, Elsevier.

B.C. Cuong, Picture fuzzy sets - First results. Part 1, Seminar “Neuro-Fuzzy Systems with

Applications”, Preprint 03/2013, May 2013, Institute of Mathematics, Hanoi, Vietnam.

B.C. Cuong , Picture fuzzy sets - First results. Part 2, Seminar “Neuro-Fuzzy Systems with

Applications”, Preprint 04/2013, June 2013, Institute of Mathematics, Hanoi Vietnam.

B.C. Cuong and V. Kreinovich , Picture Fuzzy Sets - a new concept for computational

intelligence problems , in the Proceedings of the 3rd World Congress on Information and

Communication Technologies (WICT 2013), Hanoi, Vietnam, IEEE CS, 2013, p. 1-6, ISBN 918-1-


B.C.Cuong, Picture Fuzzy Sets, Journal of Computer Science and Cybernetics, Vol.30, n.4,

, 409-420.

Glad Deschrijver, Chris Cornelis, Etienne E. Kerre, On the Representation of Intuitionistic

Fuzzy t-Norms and t-Conorms", IEEE Transactions on Fuzzy Systems, Vol. 12, No. 1, 2004, pp. 45-

J. Goguen , " L -fuzzy sets", J. Math. Anal. Appl., vol. 18, 1967, 145-174.

B.C. Cuong , P.V. Hai, Some fuzzy logic operators for picture fuzzy sets, in the Proceedings of

the 2015 IEEE International Conference on Knowledge Systems Engineering, KSE 2015, IEEE

Computer Society Publications and CPS, 2015, pp.132-137, Washington

B.C.Cuong, R.T. Ngan and B.D. Hai, An involutive picture fuzzy negation on picture fuzzy

sets and some De Morgan triples, in the Proceedings of the 2015 IEEE International Conference on

Knowledge Systems Engineering, KSE 2015, IEEE Computer Society Publications and CPS, 2015,

pp.126 -131, Washington.

B.C.Cuong, P.H.Phong and F. Smarandache, Standard Neutrosophic Soft Theory: Some First

Results, Neutrosophic Sets and Systems, vol.12, 2016, pp.80-91.

F.Smaranddache, A unifying field in logics. Neutrosophy: Neutrosophic probability, set and

logic, American Research Press, Rehoboth, 1999.

P.H. Phong, B.C. Cuong, Some intuitionistic linguistic aggregation operators, Journal of

Computer Science and Cybernetics, vol. 30, n.3, 216-226, 2014.

P.H.Phong, B.C.Cuong, Symbolic computational models for intuitionistic linguistic

information, Journal of Computer Science an Cybernetics, Vol. 32 ,n.1, 30-44, 2016.

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.

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

(1) (2017) 1-15.

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.

L.H.Son, Generalized Picture Distance Measure and Applications to Picture Fuzzy

Clustering. Applied Soft Computing 46 (2016) 284–295.

L.H. Son, Measuring Analogousness in Picture Fuzzy Sets: From Picture Distance

Measures to Picture Association, Measures. Fuzzy Optimization and Decision Making

(2017) DOI: 10.1007/s10700-016-9249-5.

L.H.Son, L DPFCM: A novel distributed picture fuzzy clustering method on picture fuzzy

sets. Expert systems with applications 42 (2015) 51-66.

P.H.Thong and L.H.Son, Picture Fuzzy Clustering : A New Computational Intelligence

Method, Soft Computing, v.20 (9) 3544-3562, 2016.

G.Wei, Picture fuzzy cross-entropy for multiple attribute decision making problems,

Journal of Business Economics and Management,vol.17(4),491-502,2016.

G.Wei, Picture 2-Type linguistic Bonferroni mean operators and their application to

multiple attribute decision making, Int. J. Fuzzy Systems, DOI 10.1007/s40815-016-0266-x

C.Wang, X.Zhou, H.Tu, S.Tao, Some geometric aggregation operators based on picture

fuzzy sets and their application in multiple attribute decision making, Italian Journal of

Pure and Applied Mathematics, n.37, 2017, pp.477-492.

S.V.Aruna Kumar, B.S. Harish, V.N.M.Aradhya, A picture fuzzy clustering approach for

brain tumor segmentation, in the Proceedings of the IEEE Second International Conference

On Cognitive Computing and Information Processing, CCIP, 2016.

B.C.Cuong, R.T.Ngan, L.C.Ngoc, Some Algebraic Properties of Picture Fuzzy t-Norms

and picture Fuzzy t – Conorms on Standard Neutrosophic Sets,

Mathematics, 2016.

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,

B.C.Cuong, R.T.Ngan, Some New De Morgan Triples of Operators in the Picture Fuzzy

Sets Theory, Seminar “Neuro-Fuzzy Systems with Applications”, Preprint 08/2017,

June 2017, Institute of Mathematics, Hanoi, Vietnam.

B.C.Cuong, A new direction of fuzzy logic,in the Proceedings of the 8th National

Conference on Funamental and Applied Information Technology Research (FAIR’8),

Hanoi, 9-10/7/2015, pp. 1-7, Natural Science and Technology Pub., Hanoi, 2015


Journal of Computer Science and Cybernetics ISSN: 1813-9663

Published by Vietnam Academy of Science and Technology