Logic

An introduction to category theory by V Sankrithi Krishnan

By V Sankrithi Krishnan

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Let: ∀x ∈ X : AL (x) = α ⇔ ∀β > α : inf [A]β > x ∧ ∀β < α : inf [A]β < x (8) ∀x ∈ X : AR (x) = α ⇔ ∀β > α : sup [A]β < x ∧ ∀β < α : sup [A]β > x (9) Then, by construction AL , AR are monotonic (not strictly in general), hence A is bi-monotonic. Corollary 3. Gaussian membership functions (and their complements) are strictly bi-monotonic. Triangular and trapezoidal membership functions (with their complements) are loosely bi-monotonic but, in general, not strictly bi-monotonic. Bi-monotonic Fuzzy Sets Lead to Optimal Fuzzy Interfaces 43 Definition 4.

Corollary 4. Let a family of multi-dimensional strictly bi-monotonic fuzzy sets be Φ = {A1 , A2 , . . , An } with Ai = Ai1 ×Ai2 ×· · ·×Aim and ∀x = (x1 , x2 , . . , xm ) : Ai (x) = Ai1 (x1 ) ⊗ Ai2 (x2 ) ⊗ · · · ⊗ Aim (xm ) where ⊗ is any t-norm. If, for each j = 1, 2, . . , m, the fuzzy interface Λj = Λj ({A1j , A2j , . . , Anj } , ≺j ) is optimal (with ≺j properly defined as in theorem 1), then the fuzzy interface Λ (Φ, ≺) is optimal, for any choice of the ordering relation ≺. 44 G. M. Fanelli, and C.

In particular, after the definition of two main classes of fuzzy sets, namely strictly bi-monotonic fuzzy sets and loosely bi-monotonic fuzzy sets, two theorems are proved. g. g. triangular and convex fuzzy sets in general) if stronger conditions hold. The theorems provide sufficient condition for optimality for a wide class of fuzzy sets, including convex fuzzy sets, so that their result can be useful in a broad range of fuzzy modeling contexts. The paper is organized as follows. In the next Section, a fuzzy interfaces and optimality condition are formally defined.

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