# Fuzzy set

Fuzzy sets are an extension of classical set theory and are used in fuzzy logic. In classical set theory the membership of elements in relation to a set is assessed in binary terms according to a crisp condition — an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in relation to a set; this is described with the aid of a membership function $\mu\ \to [0,1]$. Fuzzy sets are an extension of classical set theory since, for a certain universe, a membership function may act as an indicator function, mapping all elements to either 1 or 0, as in the classical notion.

## Definition

Specifically, a fuzzy set on a classical set Χ is defined as follows:

$\tilde{\mathit{A}}={(x,\mu_{A}(x))\mid x \in \Chi}$

The membership function μA(x) quantifies the grade of membership of the elements x to the fundamental set Χ. An element mapping to the value 0 means that the member is not included in the given set, 1 describes a fully included member. Values strictly between 0 and 1 characterize the fuzzy members.

File:Fuzzy crisp.gif
Fuzzy set and crisp set

The following holds for the functional values of the membership function μA(x)

$\mu_{A}(x)\ge0\quad\forall\quad x\in\Chi$

$\sup_{x\in X}[\mu_{A}(x)]=1$

## Applications

The fuzzy set B, where B = {(3,0.3), (4,0.7), (5,1), (6,0.4)} would be enumerated as B = {0.3/3, 0.7/4, 1/5, 0.4/6} using standard fuzzy notation. Note that any value with a membership grade of zero does not appear in the expression of the set. The standard notation for finding the membership grade of the fuzzy set B at 6 is μB(6) = 0.4.

### Fuzzy Logic

As an extension of the case of Multi-valued logic, valuations ($\mu : \mathit{V}_o \to \mathit{W}$) of propositional variables (Vo) into a set of membership degrees (W) can be thought of as membership functions mapping predicates into fuzzy sets (or more formally, into an ordered set of fuzzy pairs, called a fuzzy relation). With these valuations, many-valued logic can be extended to allow for fuzzy premises from which graded conclusions may be drawn.

This extension is sometimes called "fuzzy logic in the narrow sense" as opposed to "fuzzy logic in the wider sense," which originated in the engineering fields of automated control and knowledge engineering, and which encompasses many topics involving fuzzy sets and "approximated reasoning."

Industrial applications of fuzzy sets in the context of "fuzzy logic in the wider sence" can be found at fuzzy logic

### Fuzzy Number

A fuzzy number is a convex, normalized fuzzy set $\tilde{\mathit{A}}\subseteq\mathbb{R}$ whose membership function is at least segmentally continuous and has the functional value μA(x) = 1 at precisely one element. This can be likened to the funfair game "guess your weight," where someone guesses the contestants weight, with closer guesses being more correct, and where the guesser "wins" if they guess near enough to the contestants weight, with the actual weight being completely correct (mapping to 1 by the membership function).

### Fuzzy Interval

A fuzzy interval is a uncertain set $\tilde{\mathit{A}}\subseteq\mathbb{R}$ with a mean interval whose elements possess the membership function value μA(x) = 1. As in fuzzy numbers, the membership function must be convex, normalized, at least segmentally continuous.