Fuzzy Sets and Fuzzy Logic

Fuzzy Sets and Fuzzy Logic

In the previous chapters, a number of different methodologies for the analysis of large data sets have been discussed. Most of the approaches presented, however, assume that the data is precise. That is, they assume that we deal with exact measurements for further analysis. Historically, as reflected in classical mathematics, we commonly seek a precise and crisp description of things or events. This precision is accomplished by expressing phenomena in numeric or categorical values. But in most, if not all, real-world scenarios, we will never have totally precise values. There is always going to be a degree of uncertainty. However, classical mathematics can encounter substantial difficulties because of this fuzziness. In many real-world situations; we may say that fuzziness is reality whereas crispness or precision is simplification and idealization. The polarity between fuzziness and precision is quite a striking contradiction in the development of modern information-processing systems. One effective means of resolving the contradiction is the fuzzy-set theory, a bridge between high precision and the high complexity of fuzziness.

Fuzzy concepts derive from fuzzy phenomena that commonly occur in the real world. For example, rain is a common natural phenomenon that is difficult to describe precisely since it can rain with varying intensity, anywhere from a light shower to a torrential downpour. Since the word rain does net adequately or precisely describe the wide variations in the amount and intensity of any rain event, "rain" is considered a fuzzy phenomenon.

Often, the concepts formed in the human brain for perceiving, recognizing, and categorizing natural phenomena are also fuzzy. The boundaries ef these concepts are vague. Therefore, the judging and reasoning that emerges from them are also fuzzy. For instance, "rain" might be classified as "light rain", "moderate rain", and "heavy rain" in order to describe the degree of raining. Unfortunately, it is difficult to say when the rain is light, moderate, or heavy, because the boundaries are undefined. The concepts of "light", "moderate", and "heavy" are prime examples of fuzzy concepts themselves. To explain the principles of fuzzy sets, we will start with the basics in classical set theory.