C++ Neural Networks and Fuzzy Logic by Valluru B. Rao M&T Books, IDG Books Worldwide, Inc. ISBN: 1558515526   Pub Date: 06/01/95

Intersection and Complement of Two Fuzzy Sets

Analogously, the degree of membership of an element in the intersection of two fuzzy sets is the minimum, or the smaller value of its degree of membership individually in the two sets forming the intersection. For example, if today has 0.8 for degree of membership in the set of rainy days and 0.5 for degree of membership in the set of days of work completion, then today belongs to the set of rainy days on which work is completed to a degree of 0.5, the smaller of 0.5 and 0.8.

Recall the fuzzy sets A and B in the previous example. A = (0.9, 0.4, 0.5, 0) and B = (0.7, 0.6, 0.3, 0.8). A[cap]B, which is the intersection of the fuzzy sets A and B, is obtained by taking, in each component, the smaller of the values found in that component in A and in B. Thus A[cap]B = (0.7, 0.4, 0.3, 0).

The idea of a universal set is implicit in dealing with traditional sets. For example, if you talk of the set of married persons, the universal set is the set of all persons. Every other set you consider in that context is a subset of the universal set. We bring up this matter of universal set because when you make the complement of a traditional set A, you need to put in every element in the universal set that is not in A. The complement of a fuzzy set, however, is obtained as follows. In the case of fuzzy sets, if the degree of membership is 0.8 for a member, then that member is not in that set to a degree of 1.0 – 0.8 = 0.2. So you can set the degree of membership in the complement fuzzy set to the complement with respect to 1. If we return to the scenario of having a degree of 0.8 in the set of rainy days, then today has to have 0.2 membership degree in the set of nonrainy or clear days.

Continuing with our example of fuzzy sets A and B, and denoting the complement of A by A’, we have A’ = (0.1, 0.6, 0.5, 1) and B’ = (0.3, 0.4, 0.7, 0.2). Note that A’ [cup] B’ = (0.3, 0.6, 0.7, 1), which is also the complement of A [cap] B. You can similarly verify that the complement of A [cup] B is the same as A’ [cap] B’. Furthermore, A [cup] A’ = (0.9, 0.6, 0.5, 1) and A [cap] A’ = (0.1, 0.4, 0.5, 0), which is not a vector of zeros only, as would be the case in conventional sets. In fact, A and A’ will be equal in the sense that their fit vectors are the same, if each component in the fit vector is equal to 0.5.

Applications of Fuzzy Logic

Applications of fuzzy sets and fuzzy logic are found in many fields, including artificial intelligence, engineering, computer science, operations research, robotics, and pattern recognition. These fields are also ripe for applications for neural networks. So it seems natural that fuzziness should be introduced in neural networks themselves. Any area where humans need to indulge in making decisions, fuzzy sets can find a place, since information on which decisions are to be based may not always be complete and the reliability of the supposed values of the underlying parameters is not always certain.

Examples of Fuzzy Logic

Let us say five tasks have to be performed in a given period of time, and each task requires one person dedicated to it. Suppose there are six people capable of doing these tasks. As you have more than enough people, there is no problem in scheduling this work and getting it done. Of course who gets assigned to which task depends on some criterion, such as total time for completion, on which some optimization can be done. But suppose these six people are not necessarily available during the particular period of time in question. Suddenly, the equation is seen in less than crisp terms. The availability of the people is fuzzy-valued. Here is an example of an assignment problem where fuzzy sets can be used.

Commercial Applications

Many commercial uses of fuzzy logic exist today. A few examples are listed here:

•  A subway in Sendai, Japan uses a fuzzy controller to control a subway car. This controller has outperformed human and conventional controllers in giving a smooth ride to passengers in all terrain and external conditions.

•  Cameras and camcorders use fuzzy logic to adjust autofocus mechanisms and to cancel the jitter caused by a shaking hand.

•  Some automobiles use fuzzy logic for different control applications. Nissan has patents on fuzzy logic braking systems, transmission controls, and fuel injectors. GM uses a fuzzy transmission system in its Saturn vehicles.

•  FuziWare has developed and patented a fuzzy spreadsheet called FuziCalc that allows users to incorporate fuzziness in their data.

•  Software applications to search and match images for certain pixel regions of interest have been developed. Avian Systems has a software package called FullPixelSearch.

•  A stock market charting and research tool called SuperCharts from Omega Research, uses fuzzy logic in one of its modules to determine whether the market is bullish, bearish, or neutral.