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# What is Membership Function

Mathematical function to grade the association of a value to a set.
Published in Chapter:
Fuzzy Decision Trees
Malcolm J. Beynon (Cardiff University, UK)
DOI: 10.4018/978-1-59904-843-7.ch044
Abstract
The first (crisp) decision tree techniques were introduced in the 1960s (Hunt, Marin, & Stone, 1966), their appeal to decision makers is due in no part to their comprehensibility in classifying objects based on their attribute values (Janikow, 1998). With early techniques such as the ID3 algorithm (Quinlan, 1979), the general approach involves the repetitive partitioning of the objects in a data set through the augmentation of attributes down a tree structure from the root node, until each subset of objects is associated with the same decision class or no attribute is available for further decomposition, ending in a number of leaf nodes. This article considers the notion of decision trees in a fuzzy environment (Zadeh, 1965). The first fuzzy decision tree (FDT) reference is attributed to Chang and Pavlidis (1977), which defined a binary tree using a branch-bound-backtrack algorithm, but limited instruction on FDT construction. Later developments included fuzzy versions of crisp decision techniques, such as fuzzy ID3, and so forth (see Ichihashi, Shirai, Nagasaka, & Miyoshi, 1996; Pal & Chakraborty, 2001) and other versions (Olaru & Wehenkel, 2003).
More Results
The membership function of a fuzzy set is a generalization of the indicator function in classical sets.
A function which for each fuzzy set assigns a physical value on the abscissa to a truth value which describes to which extent (between 0 and 1) this value belongs to the set.
Membership function shows the degrees of belongingness of elements of universe of discourse say X to set A where . Simply put, it is a function that maps some set of real numbers to the interval [0, 1]. i.e. ?? A : X®[0,1].
An indicator function representing the degree of truth as an extension of valuation.
A membership function (MF) is a curve that defines how each point in the input space is mapped to a membership value (or degree of membership) between 0 and 1.
Membership of the elements of the base set in the fuzzy set.
The mathematical function that defines the degree of an element’s membership in a fuzzy set. Membership functions return a value in the range of [0,1], indicating membership degree
Membership function indicates the collection of possible values that an object may have when it is defined as a factor of another object. For example tallness can be defined as low, medium, tall. A measure, for example, 171, is defined as tallness, by the help of membership function of tallness as a factor of low, medium and tall.
It is a function that quantifies the grade of membership of a variable to a linguistic term. Reasonable functions are often piecewise linear functions, such as triangular and trapezoidal functions.
A function that quantifies the grade of membership of a variable to a linguistic term.
An indicator function representing the degree of truth as an extension of valuation
The membership function is a graphical representation of the magnitude of participation of each input. It associates weighting with each of the inputs that are processed.
Mathematical interpretation that defines multiple obedience values for variables in fuzzy sets.
Membership function is a mathematical function that defines a fuzzy set on the universe of discourse. Membership functions used in fuzzy expert systems are triangles, trapezoid and Gaussian function.
The set of elements that have a non-zero membership is called the support of the fuzzy set. The function that ties a number to each element of the universe is called the membership function.
A function used to decide the degree of truth that a variable belongs to a fuzzy set in a fuzzy logic model.
A membership function quantifies the grade of membership of a variable to a linguistic term.
A function that describes the degree of an element’s membership in a fuzzy set.
Gives the grade, or degree, of membership within the fuzzy set, of any element of the universe of discourse. The membership function maps the elements of the universe onto numerical values in the interval [0, 1].
Let, A be a set, then membership function on A is a mapping from A to real interval [0,1]. The membership function of a fuzzy set is generally denoted by . For any element in , the value of generally called the membership degree of in .
Generalization of the indicator function in classical sets. In fuzzy logic, it represents the degree of truth as an extension of valuation.
An indicator function representing the degree of truth as an extension of valuation.
A membership function shows the membership degrees of a variable to a certain set. For example, a temperature t=30° C belongs to the set hot temperature with a membership degree ?HT(30°)=0.8. The membership functions are not objective but context and subject dependent.
It is the function that gives the subjective measures for the linguistic terms.
The membership function of a fuzzy set is a generalization of the characteristic function of crisp sets. In fuzzy logic, it represents the degree of truth as an extension of valuation.