Fuzzy variables |
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Authors: | Steven Nahmias |
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Institution: | Department of Industrial Engineering Systems Management Engineering and Operations Research, University of Pittsburgh, Pittsburgh, PA 15261, U.S.A. |
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Abstract: | The purpose of this study is to explore a possible axiomatic framework from which a rigorous theory of fuzziness may be constructed. The approach we propose is analogous to the sample space concept of probability theory. A fuzzy variable is a mapping from an abstract space (called the pattern space) onto the real line. The membership function is obtained as the extension of a special type of capacity (called a scale) from the pattern space to the real line via the fuzzy variable. In essence we are proposing an entirely new definition of a fuzzy set on the line as a mapping to the line rather than on the line. The current definition of a transformation of a fuzzy set is obtained as a derived result of our model. In addition, we derive the membership function of sums and products of fuzzy sets and present an example which reinforces the credibility of our approach. |
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