Fuzzy set

Given a universe set, $LaTeX: X$, and a membership function, $LaTeX: u : X \rightarrow [0,1]$, a fuzzy set is a collection of pairs: $LaTeX: \{(x, u(x)): x \in X\}$. Often, the membership function is subscripted by the set name, say $LaTeX: u_S$. Generally, for all $LaTeX: x \in X$, $LaTeX: u_{X}(x)=1$, and $LaTeX: u_{\emptyset}(x)=0$. In the context of uncertainty, the value $LaTeX: u_{S}(x)$ is used to model the statement of how possible it is for $LaTeX: x$ to be in $LaTeX: S$. For this reason, $LaTeX: u_S$ is sometimes called the possibility function of $LaTeX: S$. What we consider the usual set (without a membership function) is called a crisp set in fuzzy mathematics.

Fuzzy set operations, say on fuzzy sets $LaTeX: S$ and $LaTeX: T$, with membership functions $LaTeX: u_S$ and $LaTeX: u_T$, resp., are defined by the following:

• Union: $LaTeX: u_{S\vee T}(x) = \max \{u_S(x), u_T(x)\}$.
• Intersection: $LaTeX: u_{S\wedge T}(x) = \min \{ u_S(x), u_T(x)\}$.
• Complement: $LaTeX: u_{~S}(x) = 1 - u_S(x)$.

One must be careful when using fuzzy sets to represent uncertainty (which is not the only type of interpretation – see fuzzy mathematical program). In particular, if $LaTeX: u_S(x) = 1/2$, its complement is also $LaTeX: 1/2$. Thus, $LaTeX: u_{S\vee \sim S}(x) = 1/2$, despite the fact that $LaTeX: S\vee \sim S = X$ (in ordinary set theory). Similarly, $LaTeX: u_{S\wedge \sim S}(x) = 1/2$, despite the fact that $LaTeX: S\wedge \sim S = \emptyset$. This illustrates the fact that $LaTeX: u_S$ need not equal $LaTeX: u_T$ even if $LaTeX: S=T$ as crisp sets.

While the fuzzy set is fundamental for fuzzy mathematical programming, other concepts in fuzzy mathematics also apply, such as fuzzy arithmetic, fuzzy graphs and fuzzy logic.