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

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*T-norm operator:*

A∩B ↔ µ_{A∩B} (x) = T (µ_{A}(x), µ_{B}(x)) = µ_{A}(x) ∧ µ_{B}(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:** T_{m}(a, b) = min (a,b) = a ∧ b

**Algebraic product:** T_{a}(a, b) = a.b

**Bounded product:** T_{b}(a, b) = 0 V (a+b-1)

**Drastic product:** T_{d}(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 ↔ µ_{A∪B} (x) = S (µ_{A}(x), µ_{B}(x)) = µ_{A}(x) ∨ µ_{B}(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:** S_{m}(a, b) = max (a,b) = a V b

**Algebraic sum:** S_{a}(a, b) = a + b – a.b

**Bounded sum:** S_{b}(a, b) = 1 ∧ (a+b)

**Drastic sum:** S_{d}(a, b) = a; b; 1 if (b=0; a=0; a, b >0)