Fuzzy T-norm and S-norm Operator

T-norm (Triangular norm) ——> Fuzzy Intersection

T-norm operator:

A∩B ↔ µA∩B (x) = T (µA(x), µB(x)) = µA(x) ∧ µB(x),     ∀ x ∈ X

where ∧ is for T-norm operator (example, min. product)

Definition of T-norm operator

A T-norm operator denoted by T(a,b) is a function mapping [0,1]×[0,1] to [0,1] that satisfies the following conditions for any a,b,c,d ∈ [0,1]

Basic requirements: T:[0,1]×[0,1]→[0,1]

Boundary: T(0, 0) = 0, T(a, 1) = T(1, a) = a
Monotonicity: T(a, b) < T(c, d) if a < c and b < d
Commutativity: T(a, b) = T(b, a)
Associativity: T(a, T(b, c)) = T(T(a, b), c)

Four examples:

Minimum: Tm(a, b) = min (a,b) = a ∧ b
Algebraic product:  Ta(a, b) = a.b
Bounded product:  Tb(a, b) = 0 V (a+b-1)
Drastic product:  Td(a, b) = a; b; 0 if (b=1; a=1; a, b <1)

Fuzzy Union: T-conorm or S-norm

S-norm operator:

A∪B ↔ µA∪B (x) = S (µA(x), µB(x)) = µA(x) ∨ µB(x),     ∀ x ∈ X

where ∨ is for S-norm operator (example, maximum)

Definition of S-norm operator

A S-norm or T-conorm operator denoted by S(a,b) is a function mapping [0,1]×[0,1] to [0,1] that satisfies the following conditions for any a,b,c,d ∈ [0,1]

Basic requirements: S:[0,1]×[0,1]→[0,1]

Boundary: S(1, 1) = 1, S(a, 0) = S(0, a) = a
Monotonicity: S(a, b) < S(c, d) if a < c and b < d
Commutativity: S(a, b) = S(b, a)
Associativity: S(a, S(b, c)) = S(S(a, b), c)

Four examples:

Maximum: Sm(a, b) = max (a,b) = a V b
Algebraic sum:  Sa(a, b) = a + b – a.b
Bounded sum:  Sb(a, b) = 1 ∧ (a+b)
Drastic sum:  Sd(a, b) = a; b; 1 if (b=0; a=0; a, b >0)

 

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