DS Hooda* Honorary Professor in Mathematics, India *Corresponding author: DS Hooda, Former PVC, Honorary Professor in Mathematics, India Submission: May 25, 2018; Published: November 01, 2018 A Survey on Trigonometric Measures of Fuzzy Information and Discrimination Introduction When proposing fuzzy set, [2] concerns were explicitly centred on their potential contribution in the domain of pattern classification, processing and communication of information, abstraction, summarization, etc. Although the claims that fuzzy sets were relevant in these areas appeared unsustainable in the early sixties, however, the future development of information science and engineering proved that these intuitions were right. The specificity of fuzzy sets is to capture the idea of partial membership. The characteristic function of a fuzzy set is often called membership function and the role of that has well been explained by Singpurwalla & Booker [3] in probability measures of fuzzy sets. A generalized theory of uncertainty has been well explained by Zadeh [4] where he remarked that uncertainty was an attribute of information. Before that the path breaking work of Shannon [5] had led to a universal acceptance of the theory that information was statistical in nature. However, a perception-based theory of probabilistic reasoning with imprecise probabilities was explained by Zadeh [4]. Taking into consideration the concept of fuzzy set, De Luca & Termini [6] suggested that corresponding to Shannon’s [5] probabilistic entropy, the measure of fuzzy information could be defined as follows: [ ] 1 ( ) ( ) log ( ) (1 ( )) log(1 ( )) , n A i A i A i A i i HA x x x x µ µ µ µ = =− + − − ∑ (1.1) Where ( ) A i x µ are the membership values? Bhandari & Pal [7] parametrically generalized (1.1) as given below: 1 1 ( ) log ( ) (1 ( )) ; 0, 1. 1 n A i A i i H A x x α α α µ µ α α α =   = + − > ≠   − ∑ (1.2) On the same lines many researchers have studied various generalized fuzzy information measures. Hooda [8] and Hooda & Bajaj [9] and many more have studied various generalized additive and non-additive fuzzy information measures. Hooda & Jain [10] characterized sub additive trigonometric measure of fuzzy information corresponding to probabilistic entropy studied by Sharma & Taneja [2]. It may be noted that trigonometric measures has its own importance in application point of view, particularly in geometry. Sine and Cosine Trigonometric Fuzzy Information Measures Most of fuzzy information measures have been defined analogous to probabilistic entropies. Hooda & Mishra [1] introduced two trigonometric fuzzy information measures which have no analogous probabilistic entropies as given below: 1 1 2 1 ( ) (1 ( )) (1) ( ) sin sin 1 (2.1) 2 2 ( ) (1 ( )) (2) ( ) cos cos 1 (2.2) 2 2 n A i A i i n A i A i i x x H A x x H A πµ π µ πµ π µ = = −   = + −     −   = + −     ∑ ∑ First of all we check the validity of the proposed measures (2.1) and (2.2). Theorem 1 The fuzzy information measure given by (2.1) is valid measure. Proof: To prove that the given measure is a valid measure, we shall show that (2.1) satisfies the four properties (P1) to (P4). (P1). ( ) 1 0 H A = if and only if A is a crisp set. Evidently, 1 1 ( ) (1 ( )) ( ) sin sin 1 0 2 2 n A i A i i x x H A πµ π µ = −   = + − =     ∑ if and only if either ( ) 0 A i x µ = or ( ) 1 0 A i x µ − = for 1, 2, , . i n = …… It implies if and only if A is a crisp set. Review Article 1/6 Copyright © All rights are reserved by DS Hooda. Volume 2 - Issue - 3 Open Access Biostatistics & Bioinformatics C CRIMSON PUBLISHERS Wings to the Research ISSN 2578-0247 Abstract In the literature of fuzzy information measures, there exist many well-known parametric and non-parametric measures with their own merits and limitations. But our main emphasis is on applications of these information measures to a variety of disciplines. It has been observed that trigonometric measure of fuzzy information measures have their own importance for application point of view particularly to geometry. In present communication the concept of fuzzy information measure is introduced with its generalizations. Some new trigonometric measures of fuzzy information due to various authors are defined and characterized. One generalized measure of fuzzy discrimination by Hooda [1] is proposed and its application in decision making is also studied.