ORIGINAL PAPER Novel distance measures for Pythagorean fuzzy sets with applications to pattern recognition problems P. A. Ejegwa 1 • J. A. Awolola 1 Received: 11 March 2019 / Accepted: 11 May 2019 Ó Springer Nature Switzerland AG 2019 Abstract Pythagorean fuzzy set (PFS) is a concept that generalizes intuitionistic fuzzy sets. The notion of PFSs is very much applicable in decision science because of its unique nature of indeterminacy. The main feature of PFSs is that it is characterized by membership degree, non-membership degree, and indeterminate degree in such a way that the sum of the square of each of the parameters is one. In this paper, we propose some novel distance measures for PFSs by incorporating the conventional parameters that describe PFSs. We provide a numerical example to illustrate the validity and applicability of the distance measures for PFSs. While analyzing the reliability of the proposed distance measures in comparison with similar distance measures for PFSs in the literature, we discover that the proposed distance measures, especially, d 5 yields the most reasonable measure. Finally, some applications of d 5 to pattern recognition problems are explicated. These novel distance measures for Pythagorean fuzzy sets could be applied in decision making of real-life problems embedded with uncertainty. Keywords Distance measure  Fuzzy set  Intuitionistic fuzzy set  Pattern recognition  Pythagorean fuzzy set 1 Introduction Zadeh (1965) proposed the concept of fuzzy sets to cope uncertainty in real-life problems. Fuzzy set theory has achieved a great success in several fields due to its ability to cope uncertainty. Fuzzy set is characterized by a mem- bership function, l which takes value from a crisp set to a unit interval, I ¼½0; 1. Many application of fuzzy sets have been carried out (see Chen et al. 2001; Chen and Tanuwijaya 2011; Chen and Chang 2011; Chen et al. 2012; Chen and Huang 2003; Lee and Chen 2008; Cheng et al. 2016; Chen and Wang 1995; Wang and Chen 2008). Out of several generalizations of fuzzy set theory for various objectives, the notion of intuitionistic fuzzy sets (IFSs) introduced by Atanassov (1983, 1986) is interesting and useful. A lot of attentions were drawn to the development of distance measures between IFSs in a quest to apply IFSs to solve many real-life problems. As such, several measures were proposed (see Hatzimichailidis et al. 2012; Szmidt and Kacprzyk 2000; Szmidt 2014; Wang and Xin 2005). Some applications of IFSs in real-life problems have been extensively researched by Davvaz and Sadrabadi (2016), Chen and Chang (2015) , Chen et al. (2016a, b), Liu and Chen (2017, 2018), Liu et al. (2017), De et al. (2001), Ejegwa et al. (2014), Ejegwa (2015), Ejegwa and Modom (2015), Ejegwa and Onasanya (2019), Szmidt and Kacpr- zyk (2001, 2004). In a pursuit to reasonably cope uncertainty in real-life problems, Yager (2013a, b) proposed a concept called Pythagorean fuzzy sets (PFSs). The theory of PFSs is a new approach to deal with vagueness more precisely in com- parison with IFSs. Albeit, the origin of PFSs emanated from IFSs of second type (IFSST) introduced by Atanassov (1989) as generalized IFSs. Some theoretical aspects of PFSs have been extensively studied (see Dick et al. 2016; Gou et al. 2016; He et al. 2016; Peng and Yang 2015). Pythagorean fuzzy set theory has attracted attentions of many scholars, and the concept has been applied to several & P. A. Ejegwa ocholohi@gmail.com J. A. Awolola remsonjay@yahoo.com 1 Department of Mathematics/Statistics/Computer Science, University of Agriculture, P.M.B. 2373 Makurdi, Nigeria 123 Granular Computing https://doi.org/10.1007/s41066-019-00176-4