Lecture 1: Introduction to Fuzzy Sets

A fuzzy set is totally characterised by a Membership function (MF). And is mathematically expressed as $S = (x, μ_{s} (x) | x \in X)$ The above equation is known as characteristic equation. Where $S$ is a fuzzy set, $μ_{s} (x)$ is a membership function (MF). A membership function basically tells how much does a element belong to a set. Finally, $X$ is called the universe of discourse.

When $X$ is discrete, $S = \sum_{x_{i} \in X} μ_{S} (x_{i}) / x_{i}$ and when $X$ is continous, $S = \int_{X} μ_{S} (x_{i}) / x_{i}$

Fuzzy set operations

Subset/containment: $A \subseteq B \Leftrightarrow μ_{A} \leq μ_{B}$
Completement: $μ_{\bar{A}} (x) = 1 - μ_{A} (x)$
Union: $C = A \cup B \Leftrightarrow μ_{C} (x) = μ_{A} (x) \lor μ_{B} (x)$
Intersection: $C = A \cap B \Leftrightarrow μ_{C} (x) = μ_{A} (x) \land μ_{B} (x)$

Fuzzy set terms

Support => $μ_{S} (x) > 0$
Core => $μ_{S} (x) = 1$
Normality => When core set is non-empty
Crossover => $μ_{S} (x) = 0.5$
Singleton => When Core(S) = Support(S)

Cartesian product

When there are two fuzzy sets $A$ and $B$ in $X$ and $Y$ , respectively. A operation called cartesian product creates a new product space $X \times Y$ for all combinations of elements in input fuzzy sets. $μ_{A \times B} (x, y) = m i n μ_{A} (x), μ_{B} (y) = μ_{A} (x) \land μ_{B} (y)$ $A \times B \Rightarrow A \land B$

Cyclindrical Extension

It is used to deproject or deaggregate values. In other words, this operation increases the dimension of a space $1 D => 2 D$

Two important notions to understand: dimension and measure.

Dimension are sets which define the space along which we can aggregate information (value of variables).
Measure is a special variable which represents the aggregated information (variables).

In fuzzy logic, if $A \subseteq X$ is a fuzzy set. Then cylindrical extension in the space $X \times Y$ is $μ_{C (A)} (x, y) := μ_{A}$

Projection

Projection can be viewed as aggregation or consolidation of information. In other words, reduces the dimension of a space $2 D => 1 D$ . $R_{y} = \int_{X} m a x_{x} μ_{R} (x, y) / y$ $R_{x} = \int_{Y} m a x_{y} μ_{R} (x, y) / x$

Extension principle

A function $f : X_{1} \times X_{2} \times \dots \times X_{n} \to Y$ maps n-dimensions in $X$ (input space) to $Y$ . Assuming that there is $A_{1}, A_{2}, \dots, A_{n}$ fuzzy sets in $X_{1}, X_{2}, \dots, X_{n}$ . The function would induce another fuzzy set in the output space $Y$ with membership function,

$μ_{B} (y) = {\begin{cases} max_{x_{1}, \dots, x_{n} \in f^{- 1} (x)} μ_{A} (x), f^{- 1} (x) \neq 0 0, o t h e r w i s e \end{cases}$

since $X = X_{1} \times X_{2} \times \dots \times X_{n}$ , [[#Cartesian product]] could be used to determine that, $μ_{A} (x) = μ_{A_{1}} (x) \land \dots m u_{A_{n}} (x)$ $μ_{A} = min_{i} μ_{A} (x)$ Substituing the above equation in the $μ_{B} (y)$ , we get that

$μ_{B} (y) = max_{(x_{1}, \dots, x_{n}) \in f^{- 1} (x)} min_{i} μ_{A} (x)$

Different dimension altering ops

Cylindrical extension => $X \to X \times Y$
Cartesian product => $X, Y \to X \times Y$
Extension principle => $X \to Y$