📈 Membership function
The Membership Function (MF) is the heart of fuzzy set theory. It determines how strongly an element belongs to a fuzzy set.
📌 Definition:
A Membership Function (μ) maps each input value to a membership degree in a fuzzy set, ranging from 0 (not a member) to 1 (fully a member).
So instead of “yes” or “no”, it says “how much” an element belongs to the fuzzy set.
🧠 Purpose of Membership Function
• To quantify fuzziness.
• To help fuzzy systems make decisions based on degrees of belonging rather than absolute truth.
📊 Mathematical Representation:
If A is a fuzzy set defined on universe X, then:
A = {(x, \mu_A(x)) \mid x \in X}
Where:
• x → input variable
• \mu_A(x) \in [0, 1] → degree of membership of x in set A
🌟 Types of Membership Functions (MFs)
| Type | Graph Shape | Use Case |
|---|---|---|
| Triangular MF | Triangle | Simple control systems |
| Trapezoidal MF | Trapezoid | Systems with range-based criteria |
| Gaussian MF | Bell-shaped curve | Smooth, natural variation (e.g., sensors) |
| Sigmoidal MF | S-curve | Smooth increasing/decreasing behavior |
| Generalized Bell MF | Flexible curve with adjustable slope | Adaptive systems |
🔺 1. Triangular MF
\mu(x) = \begin{cases} 0 & x \leq a \ \frac{x-a}{b-a} & a < x \leq b \ \frac{c-x}{c-b} & b < x \leq c \ 0 & x \geq c \end{cases}
Example:
Low Temperature (°C):
a=10, b=20, c=30 → 15°C → μ = 0.5
▭ 2. Trapezoidal MF
\mu(x) = \begin{cases} 0 & x \leq a \ \frac{x-a}{b-a} & a < x \leq b \ 1 & b < x < c \ \frac{d-x}{d-c} & c \leq x < d \ 0 & x \geq d \end{cases}
Used when a range of values have full membership (like medium speed between 40-60 km/h).
🌐 3. Gaussian MF
\mu(x) = \exp \left( -\frac{(x - c)^2}{2\sigma^2} \right)
• Smooth and symmetric
• Good for systems with gradual transitions
➕ Key Properties of MFs
| Property | Description |
|---|---|
| Normalization | Maximum μ(x) is 1 |
| Support | Range of x for which μ(x) > 0 |
| Crossover Point | Where μ(x) = 0.5 |
| Core | Values where μ(x) = 1 (full membership) |
📈 Visualization Example:
Let’s say we define three fuzzy sets for “Speed”:
• Slow: Triangular MF from 0–30 km/h
• Medium: Trapezoidal MF from 30–70 km/h
• Fast: Gaussian MF centered at 80 km/h
These MFs will overlap smoothly, allowing an input like 50 km/h to belong partially to Slow (0.2), Medium (0.8).
🔧 Real-World Example:
In a fuzzy washing machine:
• “Dirt Level” → Membership functions like Low, Medium, High
• If dirtiness = 70%, it may belong:
• 0.3 to Medium
• 0.7 to High
Then the fuzzy controller decides wash duration based on these memberships.
✍️ Exam-Ready Answer Summary:
Membership Function (MF) is a graphical representation that defines how each point in the input space is mapped to a membership value between 0 and 1. It is the foundation of fuzzy sets, helping to model vague concepts. Different types of membership functions (triangular, trapezoidal, Gaussian, sigmoidal) are used based on application needs. MFs play a key role in fuzzy reasoning and decision-making systems by allowing partial membership and smooth transitions between states.
🎯 Mnemonic for MF Types: “TTGS-B”
• T – Triangular
• T – Trapezoidal
• G – Gaussian
• S – Sigmoidal
• B – Bell-shaped