A Welcome Message from The Editor

It is with profound pleasure that we write this editorial to welcome you to the new journal, “Pakistan Biomedical Journal” (PBMJ), an interdisciplinary international journal. PBMJ has successfully completed its first volume and now its the second volume. We greatly appreciate the response of scientists who have contributed previously and are still contributing to this new journal. The subject of the journal is interesting and we try to address the health related concerns of public and improve the understandingof scientific phenomenons by researchers. Research discoveries are happening at a fast pace, in all the fields and PBMJ provides an ideal forum for exchange of scientific knowledge in terms of full length papers, surveys, reviews, case studies, letters to editor and systematic analysis. PBMJ is committed to publishing all manuscripts receiving a high recommendation from reviewers. The intention of PBMJ is to create space for generation of new knowledge, debate, collaborations among national and international scientists. Our vision is to promote research that will be helpful for knowledge sharing, new discoveries, development of critical thinking among the upcoming scholars, guidance for policy makers, awareness among the concerned community and ultimately benefitting the general population in improving health and fitness at large. It is a matter of pride for us to haveexcellent editorial board members from renowned institutes. We aim to have the best standards of quality of the published manuscripts. With every issue, we are continuously trying to improve the standards. We look forward for more exciting researches and scientific studies from all over the world. We would like to extend a very warm welcome to the readers of PBMJ and hope you will join us as authors, reviewers and editors in future.

Bell-Shaped Inequalities in Hexa-Parametric Family of Similarity Measurement

In this thesis, we study the T-transitive family of cardinality-based fuzzy similarity and inclusion measures and obtain the necessary and su cient parametric conditions to characterize the T-transitive members of the parametric family of the cardinality based similarity measure. Firstly, In crisp logic every object is similar to itself with degree of reflexivity 1, while the degree of reflexivity in fuzzy logic can be any value in the unit interval, (0; 1]. This behaviour of a fuzzy set is used to enlighten the concept of similarity and inclusion measures and discovering the relations between the parameters of the transitive members of a family of cardinality-based fuzzy measure. We present meta-theorems stating general conditions ensuring that certain inequalities for cardinalities of ordinary sets are preserved under fuzzification, when adopting a scalar approach to fuzzy set cardinality. The conditions pertain to the commutative conjunctor used for modelling fuzzy set intersection. In particular, this conjunctor should fulfil a number of Bell-type inequalities. The advantage of these meta-theorems is that repetitious calculations can be avoided. This is illustrated in the demonstration of the Łukasiewicz transitivity of fuzzified versions of the simple matching coe cient and the Jaccard coe cient, or equivalently, the triangle inequality of the corresponding dissimilarity measures. Finally, a new class of rational cardinality-based similarity measure using multiplication of cardinalities is introduced and discussed the monotonicity, boundary conditions and transitivity.