# Fuzzy formulas, with meanings and worked-example links

Subject: Intelligent Systems II

Exam: M2

Reader: https://deep-learning-study-room.vercel.app/is2/#m2/formulas/m2-formulas-v2/fuzzy

Canonical: https://deep-learning-study-room.vercel.app/read/is2/m2/formulas/m2-formulas-v2--fuzzy/

One place for the formulas, meanings, conditions and links to full worked examples.

## Fuzzy formulas

### Triangular membership



$$
\mu(x)=\max\left(0,\min\left(\frac{x-a}{b-a},\frac{c-x}{c-b}\right)\right)
$$



a=left zero, b=peak, c=right zero, with a<b<c. Locate x before choosing the rising/falling branch.

[Read the explanation and worked sums →](https://deep-learning-study-room.vercel.app/read/is2/m2/read/fuzzy-core--triangle/)

### Union and intersection



$$
\mu_{A\cup B}=\max(\mu_A,\mu_B)\quad\mu_{A\cap B}=\min(\mu_A,\mu_B)
$$



Compare memberships at the same element. These formulas use standard max–min operations.

[Read the explanation and worked sums →](https://deep-learning-study-room.vercel.app/read/is2/m2/read/fuzzy-core--set-operations/)

### Complement and difference



$$
\mu_{A^c}=1-\mu_A\quad\mu_{A\setminus B}=\min(\mu_A,1-\mu_B)
$$



A difference means A AND NOT B; it is not ordinary subtraction.

[Read the explanation and worked sums →](https://deep-learning-study-room.vercel.app/read/is2/m2/read/fuzzy-core--set-operations/)

### Rule strength, clipping and aggregation



$$
\alpha_r=\min(a_r,b_r)\quad\mu_r(z)=\min(\alpha_r,\mu_{C_r}(z))\quad\mu_{out}(z)=\max_r\mu_r(z)
$$



a,b are input degrees; α is the strength; z is a candidate output value. OR conditions use max instead.

[Read the explanation and worked sums →](https://deep-learning-study-room.vercel.app/read/is2/m2/read/fuzzy-core--rules-clipping/)

### Mean of maxima



$$
z^*=\frac{\sum_{z_i\in M}z_i}{|M|}\quad\text{one plateau }[a,b]:\ z^*=(a+b)/2
$$



M is the set of discrete output locations attaining maximum membership. The plateau shortcut is for one continuous maximum interval.

[Read the explanation and worked sums →](https://deep-learning-study-room.vercel.app/read/is2/m2/read/fuzzy-core--mom/)

### Discrete / continuous centroid



$$
z^*=\frac{\sum_i z_i\mu_i}{\sum_i\mu_i}\qquad z^*=\frac{\int z\mu(z)\,dz}{\int\mu(z)\,dz}
$$



Use actual discrete memberships or the full continuous aggregate. Denominator must be positive.

[Read the explanation and worked sums →](https://deep-learning-study-room.vercel.app/read/is2/m2/read/fuzzy-core--centroid/)

### Specified representative-weighted output



$$
z^*=\frac{\sum_r\alpha_r c_r}{\sum_r\alpha_r}
$$



c is a supplied representative such as a peak. Do not call this the exact area centroid of a clipped Mamdani aggregate without justification.

[Read the explanation and worked sums →](https://deep-learning-study-room.vercel.app/read/is2/m2/read/fuzzy-core--weighted-average/)

### Cartesian product and max–min composition



$$
R_{ij}=\min(A_i,B_j)\quad T_{ik}=\max_j\min(R_{ij},S_{jk})
$$



Relations must have a common middle domain in the same order. Min along a path, max across paths.

[Read the explanation and worked sums →](https://deep-learning-study-room.vercel.app/read/is2/m2/read/fuzzy-relations--composition/)

## Other membership formulas from the reading lessons

### Trapezoid with distinct breakpoints



$$
\mu(x)=\max\left(0,\min\left(\frac{x-a}{b-a},1,\frac{d-x}{d-c}\right)\right)
$$



For a<b≤c<d: a is the left zero, b the start of the height-1 plateau, c its end and d the right zero. Vertical shoulder edges need the specified piecewise definition. [Branch examples →](https://deep-learning-study-room.vercel.app/read/is2/m2/read/fuzzy-core--trapezoid-intuition/)

### Gaussian membership



$$
\mu(x)=e^{-(x-c)^2/(2\sigma^2)}
$$



c is the centre and σ>0 sets the width. At x=c the exponent is zero and membership is 1. [Worked exponential calculations →](https://deep-learning-study-room.vercel.app/read/is2/m2/read/fuzzy-core--smooth-memberships/)

### Generalised bell membership



$$
\mu(x)=\frac{1}{1+|(x-c)/a|^{2b}}
$$



a>0 controls width; b>0 controls steepness; c is the centre. First subtract c, divide by a, take absolute value, apply the power, add 1, then take its reciprocal.

### Alpha-cut



$$
A_\alpha=\{x:\mu_A(x)\ge\alpha\}
$$



α is the chosen membership threshold. For 0<α≤1 keep inputs meeting or exceeding it, including threshold-equality endpoints. [Discrete and triangular examples →](https://deep-learning-study-room.vercel.app/read/is2/m2/read/fuzzy-core--properties/)