A New Class of Necessity Measures and Fuzzy Rough Sets Based on Certainty Qualifications

Abstract

In this paper, we propose a new class of necessity measures which satisfy (Rl) N_A(B) > 0 ⇔ ∃_ε> 0; [A]_1-ε ⊆ (B)_ε, (R2) ∃h* ∈ (0,1); N _A(B) ≥ h* ⇔ A ⊆ B and (R3) N _A(B) = 1 ⇔ (A)₀ ⊆ [B]₁. It is shown that such a necessity measure is designed easily by level cut conditioning approach. A simple example of such a necessity measure is given. The proposed necessity measure is applied to fuzzy rough set based on certainty qualifications. It is demonstrated that the proposed necessity measure gives better upper and lower approximations of a fuzzy set than necessity measures defined by S-, R- and reciprocal R-implications.