Symmetric difference operators on fuzzy sets

Based on the properties of symmetric difference of sets, a symmetric difference operator for fuzzy sets is defined to be a continuous and associative binary operator on the closed unit interval with some boundary condition. Structures and properties of these operators are investigated in this paper. The main results are: (1) It is proved that a symmetric difference operator is determined by a continuous t-conorm and a strong negation operator on the unit interval. (2) Two models of these operators are discussed. These models are related to the solutions of certain functional equations on the unit interval. In particular, the results presented here provide a partial answer to a problem raised by Alsina, Frank and Schweizer in 2003 about functional equations on the unit interval.