Membership function (mathematics)

The membership function of a fuzzy set is a generalization of the indicator function in classical sets. In fuzzy logic, it represents the degree of truth as an extension of valuation. Degrees of truth are often confused with probabilities, although they are conceptually distinct, because fuzzy truth represents membership in vaguely defined sets, not likelihood of some event or condition.

For the universe $\mathrm {X}$ and given the membership-degree function $\mu \ \to [0,1]$

the fuzzy set A is defined as

${\tilde {\mathit {A}}}=\{(x,\mu _{A}(x))\mid x\in \mathbf {X} \}$

The membership function $\mu _{A}(x)$ quantifies the grade of membership of the elements $x$ to the fundamental set $\mathbf {X}$. The value 0 means that the member is not included in the given set, 1 describes a fully included member. The values between 0 and 1 characterize fuzzy members.

Fuzzy set and crisp set

The following holds for the functional values of the membership function $\mu _{A}(x)$

$\mu _{A}(x)\geq 0\quad \forall \quad x\in \mathbf {X}$

$\sup _{x\in X}[\mu _{A}(x)]=1$

Capacity

One application of membership functions is as capacities in decision theory.

In decision theory, a capacity is defined as a function, $\nu$ from S, the set of subsets of some set, into $[0,1]$, such that $\nu$ is set-wise monotone and is normalized (ie $\nu (\emptyset )=0,\nu (\Omega )=1$. Clearly this is a generalization of a probability measure, where the probability axiom of countability is weakened. A capacity is used as a subjective measure of the likelyhood of an event, and the "expected value" of an outcome given a certain capacity can be found by taking the Choquet integral over the capacity.

See also

• Defuzzification
• Fuzzy measure theory
• fuzzy set operations
• Rough set

External links

• Fuzzy Image Processing