Remarks on the Subtraction and Division Operations over Intuitionistic Fuzzy Sets and Interval-Valued Fuzzy Sets

Atanassov (Fuzzy Sets and Systems 20 (1986) 87-96) defined basic operations over the intuitionistic fuzzy sets. Here we shall introduce two operations, subtraction and division, derived from the deconvolution for equations using addition and multiplication operations, respectively. In addition, we shall give some conditions to examine the correctness regarding the defined operations. The remarks in this paper can be immediately expressed in terms of interval-valued fuzzy set Atanassov [1-3], [5] defined addition and multiplication operations as follows: { }, ) ( ) ( ), ( ) ( ) ( ) ( , X x x x x x x x x B A B A B A B A ∈ ⋅ ⋅ − + = + ν ν μ μ μ μ (1) { }. ) ( ) ( ) ( ) ( ), ( ) ( , X x x x x x x x x B A B A B A B A ∈ ⋅ − + ⋅ = ⋅ ν ν ν ν μ μ (2) It is easy to demonstrate the correctness of the above-mentioned operations. First, consider four non-negative real numbers ) (x A μ , ) (x A ν , ) (x B μ , and ) (x B ν , for which 1 ) ( ) ( 0 ≤ + ≤ x x A A ν μ and ) ( 0 x B μ ≤ 1 ) ( ≤ + x B ν . Since 0 ) ( 1 ≥ − x A μ and 0 ) ( 1 ≥ − x B μ , s.