π Operations on Fuzzy set
Fuzzy set theory extends classical set operations like Union, Intersection, and Complement, but instead of binary membership (0 or 1), it works on degrees of membership (0 to 1).
Letβs understand each operation step by step π
π 1. Union (OR Operation)
The membership of an element in the union of two fuzzy sets A and B is given by the maximum (max) of their individual memberships.
π’ Formula:
\mu_{A \cup B}(x) = \max[\mu_A(x), \mu_B(x)]
π‘ Example:
Let:
β’ \mu_A(x) = 0.6
β’ \mu_B(x) = 0.8
Then,
\mu_{A \cup B}(x) = \max(0.6, 0.8) = 0.8
π 2. Intersection (AND Operation)
The membership of an element in the intersection of two fuzzy sets A and B is given by the minimum (min) of their individual memberships.
π’ Formula:
\mu_{A \cap B}(x) = \min[\mu_A(x), \mu_B(x)]
π‘ Example:
Using the same values:
\mu_{A \cap B}(x) = \min(0.6, 0.8) = 0.6
π 3. Complement (NOT Operation)
The complement of a fuzzy set A represents non-membership of elements in A. It is calculated by subtracting the membership value from 1.
π’ Formula:
\mu_{A{\prime}}(x) = 1 - \mu_A(x)
π‘ Example:
\mu_{A}(x) = 0.6 β \mu_{A{\prime}}(x) = 1 - 0.6 = 0.4
π 4. Algebraic Product and Sum
| Operation | Formula | Description |
|---|---|---|
| Algebraic Product (A Γ B) | \mu_{A \cdot B}(x) = \mu_A(x) \cdot \mu_B(x) | Used in fuzzy inference systems |
| Algebraic Sum (A β B) | \mu_{A \oplus B}(x) = \mu_A(x) + \mu_B(x) - \mu_A(x) \cdot \mu_B(x) | Alternative to fuzzy union |
π 5. Bounded Difference and Sum
| Operation | Formula | Description |
|---|---|---|
| Bounded Sum | \mu_{A \oplus B}(x) = \min[1, \mu_A(x) + \mu_B(x)] | Prevents exceeding value 1 |
| Bounded Difference | \mu_{A \ominus B}(x) = \max[0, \mu_A(x) - \mu_B(x)] | Useful in fuzzy subtraction |
π 6. Drastic Union and Intersection (Extreme Case)
| Operation | Formula |
|---|---|
| Drastic Union | \mu_{A \cup B}(x) = \begin{cases} \mu_A(x) & \text{if } \mu_B(x)=0 \ \mu_B(x) & \text{if } \mu_A(x)=0 \ 1 & \text{otherwise} \end{cases} |
| Drastic Intersection | \mu_{A \cap B}(x) = \begin{cases} \mu_A(x) & \text{if } \mu_B(x)=1 \ \mu_B(x) & \text{if } \mu_A(x)=1 \ 0 & \text{otherwise} \end{cases} |
π Visual Representation (Recommended for Revision)
If youβd like, I can share a diagram comparing fuzzy set operations visually on a number line or via graph plots (Union, Intersection, Complement curves).
π Exam-Oriented Summary:
Operations on fuzzy sets are extensions of classical set operations but are based on degrees of membership. Union is modeled using the max operator, intersection using the min operator, and complement as the difference from 1. These operations allow modeling of real-world scenarios where partial memberships exist. Advanced operations like algebraic and bounded sum/product enhance fuzzy reasoning capabilities in systems like fuzzy controllers.
π― Mnemonic for Revision: βU-I-C-A-B-Dβ
β’ U β Union (Max)
β’ I β Intersection (Min)
β’ C β Complement (1 β ΞΌ)
β’ A β Algebraic (Product/Sum)
β’ B β Bounded (Sum/Diff)
β’ D β Drastic Ops