🌫️ Fuzzy Set theory
📌 Why Fuzzy Set Theory?
Traditional set theory (crisp sets) deals with binary membership—either an element belongs or does not belong to a set.
But real-world problems are rarely black-and-white.
That’s where Fuzzy Set Theory, introduced by Lotfi A. Zadeh in 1965, comes in.
🧠 What is a Fuzzy Set?
A fuzzy set is a set without sharp boundaries.
Each element in a fuzzy set has a degree of membership (from 0 to 1).
🔸 Example:
In a Crisp Set:
If Temperature > 30°C ⇒ “Hot” = Yes, else No.
In a Fuzzy Set:
Temperature = 28°C → Membership in “Hot” = 0.6 (60% hot)
🔢 Mathematical Representation
A Fuzzy Set A in Universe of Discourse X is defined as:
A = { (x, \mu_A(x)) \mid x \in X }
Where:
• x → element of X
• \mu_A(x) → Membership Function of set A, value ∈ [0, 1]
📊 Membership Function (MF) – The Core of Fuzzy Logic
It maps each element to a degree of belonging.
Common MF Types:
| Type | Description | Use Case |
|---|---|---|
| Triangular MF | Simple & computationally light | Basic control systems |
| Trapezoidal MF | Flat top region | Washing machines |
| Gaussian MF | Smooth bell-shaped curve | Medical & NLP applications |
| Sigmoidal MF | S-curve | Smooth transition scenarios |
🎯 Example: Triangular MF for Temperature
Temp (°C): 20 25 30
MF Value : 0.0 1.0 0.0
Here, 25°C is fully “Warm”, but 22°C might be 0.4 warm, etc.
🔍 Fuzzy Set Operations (Similar to Crisp Set but with Degrees)
| Operation | Formula | Example |
|---|---|---|
| Union (A ∪ B) | max[μA(x), μB(x)] | Combine possibilities |
| Intersection (A ∩ B) | min[μA(x), μB(x)] | Commonality |
| Complement (A’) | 1 – μA(x) | Degree of non-membership |
⚖️ Crisp Set vs Fuzzy Set (Quick Comparison)
| Criteria | Crisp Set | Fuzzy Set |
|---|---|---|
| Membership | 0 or 1 | 0 to 1 (partial) |
| Boundary | Sharp | Flexible/Gradual |
| Example | Adult = age ≥18 | Adult = age with gradual membership from 16–22 |
💡 Real-World Applications of Fuzzy Sets
| Domain | Application |
|---|---|
| Home Automation | Fan/AC Speed Control based on Fuzzy Temperature |
| Healthcare | Fuzzy diagnosis for symptom analysis |
| NLP | Fuzzy similarity in word meanings |
| Image Processing | Fuzzy edge detection and segmentation |
✍️ Exam Answer Format:
Fuzzy Set Theory extends classical set theory by allowing partial membership of elements, defined by a membership function in the range [0, 1]. It enables modeling of vague or imprecise concepts such as “hot”, “tall”, or “fast”. Operations such as union, intersection, and complement are applied using min-max principles. Fuzzy Set Theory forms the backbone of Fuzzy Logic Systems widely used in real-world decision-making applications.
📌 Mnemonic for Revision: “MOMI”
• M – Membership Function
• O – Operations (Union, Intersection, Complement)
• M – Mathematical Representation
• I – Imprecise Reasoning