🔗 Fuzzy Relation
📌 What is a Fuzzy Relation?
A Fuzzy Relation is an extension of a crisp relation in classical set theory.
In classical logic, relations between elements are binary (either exist or not).
In Fuzzy Logic, relations are graded or fuzzy, where the degree of association between elements varies between 0 and 1.
🧠 Formal Definition:
Let there be two universes of discourse:
• X and Y
Then a fuzzy relation R from X to Y is defined as:
R = { ((x, y), \mu_R(x, y)) \mid x \in X, y \in Y }
Where:
• \mu_R(x, y) \in [0, 1] denotes the degree of relationship between element x in set X and y in set Y.
📊 Matrix Representation of Fuzzy Relation
Fuzzy Relations are often represented as a membership matrix, where:
• Rows represent elements of set X
• Columns represent elements of set Y
• Entries show \mu_R(x, y) values
Example:
Let X = {x1, x2}, Y = {y1, y2, y3}
Fuzzy Relation Matrix R(X→Y):
R = \begin{bmatrix} 0.6 & 0.9 & 0.3 \ 0.2 & 0.8 & 0.5 \end{bmatrix}
• This means:
\mu_R(x1, y1) = 0.6, \mu_R(x2, y2) = 0.8, etc.
🔄 Fuzzy Relations on Same Set (Fuzzy Relation on X×X)
A fuzzy relation on the same set (e.g., similarity between students, distances between cities) is called a fuzzy compatibility relation.
📌 Important Fuzzy Relation Operations
| Operation | Description | Formula |
|---|---|---|
| Max-Min Composition | Combines two fuzzy relations | \mu_{R \circ S}(x, z) = \max_y [\min(\mu_R(x,y), \mu_S(y,z))] |
| Max-Product Composition | Alternative to max-min | \mu_{R \circ S}(x, z) = \max_y [\mu_R(x,y) \cdot \mu_S(y,z)] |
| Inverse of Relation (R⁻¹) | Swaps the relation direction | \mu_{R^{-1}}(x, y) = \mu_R(y, x) |
| Reflexive Relation | \mu(x, x) = 1 for all x | |
| Symmetric Relation | \mu(x, y) = \mu(y, x) | |
| Transitive Relation | \mu(x, z) \geq \min[\mu(x, y), \mu(y, z)] |
🔍 Real-Life Application Examples of Fuzzy Relations
| Domain | Application |
|---|---|
| E-commerce | Relationship between customers and products (recommendation strength) |
| Healthcare | Degree of similarity between symptoms and diseases |
| Smart Home | Relationship between temperature level and fan speed |
| Image Processing | Fuzzy similarity between pixels |
✍️ Exam-Ready Summary:
A fuzzy relation is a generalization of a crisp relation where the degree of association between pairs of elements is expressed as a value between 0 and 1. It is represented by a membership function and often visualized using a matrix. Fuzzy relation operations like max-min and max-product composition are used to combine or derive new relations. These concepts are fundamental in building fuzzy inference systems, clustering, and expert systems.
🎯 Mnemonic for Quick Recall: “MIRTS”
• M – Matrix Representation
• I – Inverse Relation
• R – Reflexivity & Symmetry
• T – Transitivity
• S – Composition (Max-Min / Max-Product)