# 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
$$R = {((x, y), \mu_{R}(x,y))| (x,y) \in X \times Y}$$

#### 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

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

### 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:>Ageis 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

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