The following are explanations of the mathematics underlying the various models discussed before. Several sections discuss research issues without any mathematical treatment. Where math is presented, it is just enough to indicate the gist, and is certainly not a full and formal treatment, which would take many more pages than are allocated here. The words in many cases follow previous writings by the author of this book, some bearing the author’s own copyright and some in previous Wiley publications.

In fuzzy logic, a descriptive word (i.e., tall or short or fast or clever) has a numerical degree of truthfulness or applicability as applied to some person or thing. The truthfulness or relevance number is called *membership*, and it ranges from 0 to 1 (see Figure A.1). In the context of a college basketball player, his 6′6″ height may be 0.8 for the fuzzy term *very tall* and 0.3 for the term *moderately tall* on a hypothetical relevance scale. Grades may be a second consideration for his coach when selecting the team, who might want to distinguish brilliant players from those who are just average. The subject player might have mediocre C grades, and so have a 0.1 for *brilliant* and a 0.9 for *average*. Figure A.1 shows a plot of the hypothetical membership functions for the four fuzzy variables labeled, which can be applied to our subject basketball player.

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