A Fuzzy Calculator

How fuzzy arithmetic works

You can use fuzzy numbers for fuzzy arithmetic. This can be done by the application of Zadeh's extension principle.
  • In the cartesian product of two fuzzy numbers A and B you take the MINIMUM of the grades of membership of the two corresponding sub-numbers ai and bi that are operated on, to determine the grade of membership of the new sub-number ci resulting from that operation.

  • Then you take the MAXIMUM of the grades of membership of the sub-numbers with the same numerical value ci to determine the grade of membership of the sub-number ci of the new fuzzy number C.

    In short it's the "MAX of MIN's".

With the use of this principle I made a simple fuzzy calculator. There are two fuzzy numbers on it: A and B. In the default mode A is a "fuzzy 2" with boundaries 1 and 3 and B is a "fuzzy 4" with boundaries 3 and 5. Below each sub-number of a fuzzy number its grade of membership is shown. The result of an arithmetic operation is shown in Fuzzy C, which is an approximation of a new fuzzy number.
The main purpose of this fuzzy calculator is to show how the extension principle works.
Therefore I made the grades of membership of the lower and upper bounderies of fuzzy A and fuzzy B in the default mode not completely zero.
I let them approximate zero all slightly different from each other in order to make them traceable in the handling of the extension principle. By clicking on "Cartesian Productspace" in the fuzzy calculator you can make this cartesian productspace visible and see the new sub-numbers and the grade of membership assigned to each of them.
You can change the sub-numbers of fuzzy A and fuzzy B in the fuzzy calculator to make new fuzzy numbers and new computations. Also you can change the grades of membership. Always keep in mind that these grades do belong to the closed interval [0, 1]. Below, there also is a link to some diagrams of arithmetical operations on the default fuzzy numbers of the fuzzy calculator. It should be stressed that the diagrams of the results (in blue) are approximations.

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