Fuzzy Set Cross Product
Introduction
A fuzzy set is an extension of a classical set where elements have a membership degree in the range ([0,1]). Given two fuzzy sets A and B, their cross product is defined as a matrix where each element is computed using a T-norm (Triangular Norm), which generalizes intersection operations in fuzzy logic.
Notation
- Let A and B be two fuzzy sets: $$ A = { (a_1, \mu_A(a_1)), (a_2, \mu_A(a_2)), …, (a_m, \mu_A(a_m)) } $$ $$ B = { (b_1, \mu_B(b_1)), (b_2, \mu_B(b_2)), …, (b_n, \mu_B(b_n)) } $$
- Their cross product is given by: $$ A \times B = \sum T(\mu_A(a), \mu_B(b)) $$ where T(a, b) represents a T-norm function.
T-Norm Functions
There are different ways to define the T-norm function:
- Minimum T-norm (Mamdani’s Rule): $$ T(a, b) = \min(a, b) $$
- Algebraic Product T-norm (Larsen’s Rule): $$ T(a, b) = a \cdot b $$
- Bounded Product T-norm: $$ T(a, b) = \max(0, a + b - 1) $$
- Drastic Product T-norm: $$ T(a, b) = \begin{cases} a, & \text{if } b = 1 \ b, & \text{if } a = 1 \ 0, & \text{if } a < 1 \text{ and } b < 1 \end{cases} $$
Example Calculation
Consider two fuzzy sets:
$$ A = { (x_1, 0.2), (x_2, 0.8) } $$ $$ B = { (y_1, 0.5), (y_2, 0.7) } $$
The cross product matrix under different T-norms is:
Mamdani’s Minimum T-norm
| A \ B | 0.5 | 0.7 |
|---|---|---|
| 0.2 | 0.2 | 0.2 |
| 0.8 | 0.5 | 0.7 |
Algebraic Product T-norm
| A \ B | 0.5 | 0.7 |
|---|---|---|
| 0.2 | 0.1 | 0.14 |
| 0.8 | 0.4 | 0.56 |
Bounded Product T-norm
| A \ B | 0.5 | 0.7 |
|---|---|---|
| 0.2 | 0 | 0 |
| 0.8 | 0.3 | 0.5 |
Drastic Product T-norm
| A \ B | 0.5 | 0.7 |
|---|---|---|
| 0.2 | 0 | 0 |
| 0.8 | 0 | 0 |
(For drastic product, all values are 0 unless either A or B contains a membership value of 1.)
Bounded Product Operator
The Bounded Product of two fuzzy sets ( A ) and ( B ) is defined as:
$$ R*{bp} = A \times B = \int*{x,y} M_A(x) \odot M_B(y) , (x,y) $$
Expanding further:
$$ R*{bp} = \int*{x,y} 0 \vee \left( M_A(x) + M_B(y) - 1 \right) $$
or simply:
$$ t_{bp} = 0 \vee (a + b - 1) $$
Drastic Product Operator
The Drastic Product of two fuzzy sets ( A ) and ( B ) is defined as:
$$ R*{dp} = A \times B = \int*{x,y} M_A(y) \odot M_B(y) , (x,y) $$
or,
$$ t_{dp}(a, b) = \begin{cases} a, & \text{if } b = 1 \ b, & \text{if } a = 1 \ 0, & \text{otherwise} \end{cases} $$
Interpretation of $ A \rightarrow B $
There are three main ways to interpret the implication:
1. Material Implication
$$ R: A \rightarrow B = \bar{A} \cup B $$
2. Propositional Calculus
$$ R: A \rightarrow B = \bar{A} \cup (A \cap B) $$
3. Extended Propositional Calculus
$$ R: A \rightarrow B = (\bar{A} \cap B) \cup B $$
Fuzzy Implication Functions
Given the above interpretations, there are multiple fuzzy implication functions commonly used in fuzzy rule-based systems.
Zadeh’s Arithmetic Rule
$$ R*{za} = \bar{A} \cup B = \int*{x,y} 1 \wedge (1 - M_A(x) + M_B(y)) $$
or
$$ t_{za} = 1 \wedge (1 - a + b) $$
Zadeh’s Max-Min Rule
$$ R*{mm} = \bar{A} \cup (A \cap B) = \int*{x,y} (1 - M_A(x)) \vee (M_A(x) \wedge M_B(y)) $$
or
$$ t_{mm}(a, b) = (1 - a) \vee (a \wedge b) $$
Boolean Fuzzy Rule
$$ R_{bf} = \bar{A} \cup B $$
or
$$ t_{bf}(a, b) = (1 - a) \vee b $$
Goguen’s Fuzzy Rule
$$ R*{gf} = \int*{x,y} MA(x) \times M_B(y) $$ where, $$ t{gf}(a, b) = \begin{cases} 1, & \text{if } a \leq b \ b/a, & \text{if } a > b \end{cases} $$