A fuzzy subset F of a set Y can be defined as a set of ordered pairs, each with the first element from S, and the second element from the interval [0,1], with exactly one ordered pair present for each element of S. This defines a mapping between elements of the set S and values in the interval [0,1]. The value zero is still used to represent complete non-membership, the value one is used to represent complete membership, and values in between are used to represent intermediate degrees of membership. The set S is referred to as the universe of discourse for the fuzzy subset F. Frequently, the mapping is described as a function, the membership function of F. The degree to which the statement x is in F is true is determined by finding the ordered pair whose first element is x. The degree of truth of the statement is the second element of the ordered pair. To understand this graphically, lets use a Zadeh example that measures tallness. In this case the set S (the universe of discourse) is the set of people. We define a fuzzy subset TALL, which will answer the question "to what degree is person x tall?" Zadeh describes TALL as a linguistic variable, which represents our cognitive category of "tallness". To each person in the universe of discourse, we have to assign a degree of membership in the fuzzy subset TALL. The easiest way to do this is with a membership function based on the person's height. The following figure is how to define this using a graph. Fig. 1 tall(x) = { 0, if height(x) * 5 ft., (height(x)-5ft.)/2ft., if 5 ft. *= height (x) *= 7 ft., 1, if height(x) * 7 ft. } A graph of this looks like: 1.0 + +------------------- | / | / 0.5 + / | / | / 0.0 +-------------+-----+------------------- | | 5.0 7.0 height, ft. -* Person Height degree of tallness -------------------------------------- Billy 3' 2" 0.00 Yoke 5' 5" 0.21 Drew 5' 9" 0.38 Erik 5' 10" 0.42 Mark 6' 1" 0.54 Kareem 7' 2" 1.00 The application of fuzzy logic allows us to go beyond the simple categories of "tall" and "not tall" and apply a real world understanding of other outside variables that are always included when considering humanistic characteristics. If someone were to ask "how tall is Shaq?", you probably wouldn't say "tall", you would probably add a descriptive variable like "really tall". Fuzzy logic is a way of measuring more discrete variables for further understanding. Fuzzy logic is used directly in very few applications. One good example of its application is the Sony Palmtop. This device is one of the new handheld messaging systems that have become very popular of late. With it, you can use a computer light pen to write words on the screen. The computer then uses a form of fuzzy

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