Intelligent Systems II · M2 · formulas

Fuzzy formulas, with meanings and worked-example links

All M2 formulas

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

Fuzzy formulas

\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.

\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.

\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.

\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.

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.

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.

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.

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.