Angular Fuzzy Sets

Angular fuzzy sets differ from standard fuzzy sets merely in their coordinate description. Angular fuzzy sets are explained on a universe of angles; thus there are repeating shapes for every 2 π cycles. In mostly applications angular fuzzy sets are utilized in quantitative description of linguistic variables termed as truth values. We can suggest that the variable "truth" is no different from some other linguistic variable in that this can be described by a fuzzy set. While a certain position has a membership value of 1 this is said to be true, and while position has membership value 0 this is said to be false; values among 0 and 1 corresponds to a proposition being partially true.

We will describe angular fuzzy sets with an example in environmental engineering. Assume we measure the pH value of some water samples collected from a polluted pond and provide these pH readings linguistic labels, as like: very basic, fairly acidic to communicate qualitative information to that the pollution. This is well known that a natural solution has a pH of 7. The linguistic terms can be built in a way such a "neutral" (N) solution corresponds to θ = 0 rad, "absolutely basic" or AB corresponds to θ= ∏/2 rad, and "absolutely acidic" or AA corresponds to θ = -∏/2 rad. Levels of pH in between 7 and 0 are labeled as extremely acidic or VA, acidic or A, fairly acidic or FA, and so on such are represented in between θ = 0 and θ =- π/2.

By using these linguistic labels for pH, the model of the angular fuzzy set is shown in following diagram.

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                                        Diagram: Angular Fuzzy Sets Defined on the Unit