Abstract
Here, we study two formulas that define the symmetric difference operators of fuzzy sets. In fuzzy logic, we call this an exclusive or operator. An interesting open problem was to find a certain class of operator systems where the two well-known definitions of the symmetric difference operator are equivalent, and it is associative. Here, we state a functional equation which characterizes the equivalence of these two forms. We present a general class of operators which fulfills the associativity requirement and we show that the two definitions are equivalent. We generalize the symmetric difference operator to the n-variable case; and based on the latter the weighted form can be obtained. We will give concrete forms of these operators and we will also show that they have different types of transitivity properties.
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