The X-μ Approach: In Theory and Practice
Criticisms exist for fuzzy set theory which do not reside in classical (or “crisp”) set theory, some such issues exhibited by fuzzy set theory regard the law of excluded middle and law of contradiction. There are also additional complexities in fuzzy set theory for monotonicity, order and cardinality. Fuzzy set applications either avoid these issues, or use them to their advantage. The X-μ approach, however, attempts to solve some of these issues through analysis of inverse fuzzy membership functions. Through the inverse fuzzy membership function it is possible to computationally calculate classical set operations over an entire fuzzy membership. This paper firstly explores how the X-μ approach compares to both classical/crisp set theory and conventional fuzzy set theory, and explores how the problems regarding the laws of excluded middle and contradiction might be solved using X-μ. Finally the approach is implemented and applied to an area of big data over the world-wide-web, using movie ratings data.
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