An introduction to fuzzy math

Fuzzy math differs from conventional math primarily in the area of set theory. For example in a conventional AND statement both statements must be true for the statement to be true. However, in fuzzy logic statements are not always true or false, they merely have varying levels of confidence. The table below exibits the differences in how an AND statement is applied in conventional and fuzzy systems. As can clearly be seen the AND statement in a fuzzy system is merely the minimum confidence value of the two values.

 conventional fuzzy A B A AND B Fuzzy A Fuzzy B A AND B 1 1 0 0 1 0 1 0 1 0 0 0 .700 .345 .000 .985 1.000 .625 .453 .245 .700 .345 .000 .245

The OR statement is also different in a fuzzy system as opposed to a conventional system. Once again the table below demonstrates how an OR statement functions in both a conventional and fuzzy system. The table demonstrates that the OR statement returns the maximum value of its two operatives. Thus, the OR is essentially the opposite of the AND statement, under a fuzzy system.

 conventional fuzzy A B A AND B Fuzzy A Fuzzy B A AND B 1 1 0 0 1 0 1 0 1 1 1 0 .700 .345 .000 .985 1.000 .625 .453 .245 1.000 .625 .453 .985

The concept of the fuzzy set is also key to many applications of fuzzy logic. A set in conventional mathematics is very simple. A set is merely a collection of numbers or things. However, in fuzzy math the definition of a set is slightly different. A fuzzy set is a list of values, which describes to what degree an object belongs to a variety of characteristics. The fuzzy set listed below is an example of how a variety of temperatures might be classified.

 temp. in Farenheit very cold cold cool moderate warm hot brain baking 20 30 40 50 60 70 80 90 100 110 .850 .275 .100 .000 .000 .000 .000 .000 .000 .000 .150 .450 .400 .300 .000 .000 .000 .000 .000 .000 .000 .275 .400 .500 .000 .000 .000 .000 .000 .000 .000 .000 .100 .300 .300 .100 .000 .000 .000 .000 .000 .000 .000 .000 .500 .800 .400 .100 .000 .000 .000 .000 .000 .000 .200 .100 .500 .700 .400 .100 .000 .000 .000 .000 .000 .000 .100 .200 .600 .900