Fuzzy Logic Notes

A fuzzy set can be defined by weights and their corresponding values. For example:

$$A = 0.1/x1+0.4/x3 +1.0/x4$$

$$B = 0.3/x1 + 0.2/x2 +0.6/x3+0.2/x4$$

Here, the first term in $A$ can be viewed as a wight of $0.1$ at point $x1$

These sets can be operated on using Zadeh's operators:

  • $A^c(x)=1-A(x)$
  • $(A\cup B)(x)=max \{ A(x),B(X) \}$
  • $(A\cap B)(x)=min\{ A(x),B(x) \}$

So for example if we wanted to find the union (indicated by the $\cup$ symbol) of $A$ and $B$,

$$A\cup B = max \{ 0.1,0.3 \} /x1 + max \{ 0.0, 0.2 \} /x2 + max \{ 0.4,0.6 \} /x3 + max \{ 1.0,0.2 \} /x4$$

$$A\cup B = 0.3/x1 + 0.2/x2 + 0.6/x3 + 1.0/x4$$

And the intersection is:

$$A\cap B = min \{ 0.1,0.3 \} /x1 + min \{ 0.0, 0.2 \} /x2 + min \{ 0.4,0.6 \} /x3 + min \{ 1.0,0.2 \} /x4$$

$$A\cup B = 0.1/x1 + 0.0/x2 + 0.4/x3 + 0.2/x4$$

These are examples of T-Norms and T-Conorms. T-Norms and T-Conorms are basic algebraic operators that work on fuzzy sets.