School of Mathematics and Statistics

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Browsing School of Mathematics and Statistics by Subject "(S,N)-implication"
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    Investigations of fuzzy implications satisfying generalized hypothetical syllogism
    ( 2017-09-15) Vemuri, Nageswara Rao
    Generalized hypothetical syllogism (GHS) is one of the fuzzy inference rules that plays a central role in the selection of suitable fuzzy implications for a specific task. Because of the important role played by the classical hypothetical syllogism, in this work, we attempt to investigate the fuzzy implications that do satisfy (GHS). Due to the variety of fuzzy implications and the complexity of the functional equation, we restrict our investigations of fuzzy implications only for those which come from some well known families of fuzzy implications, viz., (S,N)-, R-, QL-, and Yager's families of f-, g-implications. Since there do not exist many fuzzy implications satisfying (GHS) from these families of fuzzy implications, we propose some classes of fuzzy implications and show that every element from these classes satisfies (GHS). Finally, we examine the preservation of (GHS) by some important generating methods of fuzzy implications that exist in the literature.
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    Mutually exchangeable fuzzy implications
    ( 2015-10-01) Vemuri, Nageswara Rao
    Recently, Vemuri and Jayaram (2012) have proposed a novel generating method of fuzzy implications, called the composition. Further, they have also proposed Mutual Exchangeability (ME), a generalization of the Exchange Principle (EP) to a pair of fuzzy implications and have shown that (ME) plays a central role in the preservation of basic properties, functional equations and families of fuzzy implications w.r.t. the composition (Vemuri and Jayaram, 2014). Due to the important role played by (ME), in this work, we investigate the pairs (I,J) of fuzzy implications that satisfy (ME). Towards this, we show first that there exist pairs (I,J) of fuzzy implications that satisfy (ME) and determine some necessary conditions on such fuzzy implications. Following this, for a given I∈I, we find the set JI of all fuzzy implications J such that the pair (I,J) satisfies (ME). Keeping in view the variety of fuzzy implications and the complexity of the functional equation (ME), we restrict our investigations to four important families of fuzzy implications, namely, (S,N)-, R-, f- and g-implications. Further, we discuss a generalization of the Cauchy multiplicative equation, whose solutions help us in obtaining the set JI for some families of fuzzy implications.