Lecture 2: Fuzzy Relations & Compositions

Fuzzy Relations

Binary relation: When there are two universe of discourse $X \times Y$ , then a fuzzy relation is defined as $$R = {((x, y), \mu_{R}(x,y))| (x,y) \in X \times Y}$$

Fuzzy relations are just multi-dimensional fuzzy sets.

Composition

When there are two fuzzy relations, defined $R_1 = X \times Y$ and $R_2 = Y \times Z$. Then a new fuzzy relation can be formed by performing projection on to the common space. $$R:= R_1 \circ R_2 \subseteq X \times Z$$ $$\mu_R(x, z) = \max_y \min[\mu_{R_1}(x, y), \mu_{R_2}(y,z)]$$ $$R_1 \circ R_2 = \vee_y[\mu_{R_1}(x, y) \wedge \mu_{R_2}(y,z)]$$

Max-min composition

$$R_1 \circ R_2 = \vee_y[\mu_{R_1}(x, y) \wedge \mu_{R_2}(y,z)]$$

Max-product composition

$$R_1 \circ R_2 = \vee_y[\mu_{R_1}(x, y) \cdot \mu_{R_2}(y,z)]$$

Set operations are performed in the same dimension. On the other hand, set compositions are performed in different dimensions.

Linguistic variables and values

Principle of incompatibility

As the complexity of the system increases, our ability to make precise and yet significant statements about its behaviour diminishes until a threshold is reached.

Beyond the threshold, precision and significance become almost mutually exclusive characteristics.

Therefore, Zadeh proposed an approach in an approximate manner, to summarise information and express it in terms of fuzzy sets.

A linguistic variable is a quintuple $(x, T(x), X, G, M)$

$x$ => Name of the variable
$T(x)$ => Term set consisting of linguistic values or terms
$X$ => Universe of discourse
$G$ => Syntactic rule, which generates terms in T(x)
$M$ => Semantic rule, which maps each linguistic value $T(x)$ to a fuzzy set in $X$

Example: > Age is linguistic variable (Note: name of the variable is “Age”)

T(Age) = {young, old, …, very old, not very young}

$X = [0, 100]$

Operations on linguistic variables

a. Exponential

$$ A^k = \int\limits_{X} [\mu_A(x)]^k/x $$

b. Concentration

$$ CON(A) = A^2 $$

c. Dilation

$$DIL(A) = A^{0.5}$$

d. Contrast Intensification

$$ INT(A) = \begin{cases} 2A^2, 0\leq u_A(x) \leq 0.5\ \neg2(\neg A^2), 0.5 < u_A(x) \leq 1 \end{cases} $$