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Topic: Membership function (mathematics)

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	Fuzzy set - Wikipedia, the free encyclopedia
		A fuzzy set is characterized by a membership function, which maps the members of the universe into the unit interval [0,1].
		The value 0 means that the member is not included in the given set, 1 describes a fully included member (this behaviour corresponds to the indicator function of classical sets).
		Note that any value with a membership grade of zero does not appear in the expression of the set.
	en.wikipedia.org /wiki/Fuzzy_set (185 words)

	Mathematics (Site not responding. Last check: 2007-11-05)
		Mathematics is the study of relationships among quantities, magnitudes and properties.
		The techniques of mathematics may be applied to a wide array of problems, such as the design and analysis of experiments, statistics and data analysis, mathematical modeling and operations research.
		The Department of Mathematics, recognizing the central position of mathematics in traditional liberal studies, provides a rigorous foundation in pure as well as applied mathematics, equipping students for further study and preparing them to use their mathematical skills in many different employment arenas.
	catalog.aucegypt.edu /undergrad/fields/math/math.html (1034 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		For triangular membership function, the minimum and maximum utility values of each level among five experts are the two bottom points, and the mean of the five utility values is considered to be the top point, the membership grades of 1.
		Membership Function for Each Weight Although all attributes are quite different, they can be compared according to their contributions to the final cost.
		In the membership function of weights, the membership grade of the weights from the GA, instead of the average of weight values given experts (see equation 3.6), is equal to 1.
	ww2.cs.fsu.edu /~wankzhao/costpaper.doc (4419 words)

	Engineering Economist, The: Fuzzy cash flow analysis using present worth criterion (Site not responding. Last check: 2007-11-05)
		The membership function may be viewed as an opinion poll of human thought.
		Thus, the membership can be explained as the possibility of an interest fate being considered as a "high annual interest rate." A rational mapping from interest rate to its degree of membership about the fuzzy set "high annual interest rate" is depicted in Figure 2.
		The membership function is a mapping from any given x to its corresponding degree of membership.
	www.findarticles.com /p/articles/mi_qa3621/is_199401/ai_n8710561 (1421 words)

	[No title]
		However, in fuzzy terms the two membership functions would read something like "Bob is very smart" and "Bob is very tall." If we presume for the sake of argument that "very" is a stronger term than "quite," and that we would correlate "quite" with the value 0.81, then the semantic difference becomes obvious.
		Fuzzy theorists argue that a sting of low membership grades should not produce a high membership gradel instead, the limit of the resulting membership grade should be the strongest membership value in the collection.
		The mathematics generated by these theories is consistent, and fuzzy logic may be a generalization of classic logic.
	www.ortech-engr.com /fuzzy/tutor.txt (2691 words)

	Anglicans Online \| The online centre of the Anglican / Episcopal world
		There is a whole branch of mathematics devoted to the study of inclusion, exclusion, and membership.
		If there is such a thing as a communion, then mathematically it is defined by a 'membership function', which gives a yes-or-no answer as to whether or not something is a member.
		There is a proof technique in mathematics called reductio ad absurdum, in which you prove a premise by showing that, were it false, a contradiction is inevitable.
	morgue.anglicansonline.org /030720 (970 words)

	An Introduction To FUzzy Control Systems
		Membership functions are a useful and integral part of fuzzy control systems.
		Instead, it is often the case that membership functions are composed of straight lines - because the computation of the membership value is much simpler in that case.
		For the composite function, the moment of the function is the integral of the function multiplied by the variable.
	www.facstaff.bucknell.edu /mastascu/eControlHTML/Fuzzy/Fuzzy1.html (3386 words)

	Fuzzy Geostat
		The membership function is increasing towards the peak and decreasing away from it.
		This determination could be explained in Fuzzy Logic words as the grade of membership (between 0 and 1) of the probability to exceed certain threshold, and then applying this function as the fuzzy boundary of being contaminated or not.
		It was intended to show that IK results could be seen as grades of membership under the concept of exceeding certain threshold, and also that such membership once transformed (or modeled) into a function can be used to interact with other membership function using logical operators and fuzzy controls.
	www.eng.vt.edu /eng/mining/nietofolder/fuzzy/fuzzy.htm (2673 words)

	GPS World - Fuzzy Logic and GPS
		Figure 1 Membership functions of different fuzzy sets: "Color is green" (top), "T is approximately between -1 and 1" (center), "Elevation is low", "Elevation is medium", and "Elevation is high" (bottom).
		A fuzzy set F˜ is defined by its membership function µF˜ which assigns a value between 0 and 1 to each possible member X: The value of this function is called the degree of membership.
		Since each fuzzy set is directly related to its membership function (MF), it may be visualized by a plot of this membership function.
	www.gpsworld.com /gpsworld/article/articleDetail.jsp?id=49880 (1310 words)

	Profile of Professor
		According to the Handbook of Mathematics and computational science, this kind of logic is known as classical logic where a statement can only have two truth values (true=1 or false=0).
		In fuzzy set, the members of the set are allowed to have membership value indicated by a value between 0 and 1.
		This read as a membership function maps every element of the universe of discourse X to the interval [0,1].
	www.sci.brooklyn.cuny.edu /~kopec/cis718/VICTOR3.htm (941 words)

	Fuzzy Logic Introduction
		The membership function is a graphical representation of the magnitude of participation of each input.
		The rules use the input membership values as weighting factors to determine their influence on the fuzzy output sets of the final output conclusion.
		The membership function, operating in this case on the fuzzy set of interferometric coherence g, returns a value between 0.0 and 1.0.
	epsilon.nought.de /tutorials/fuzzy/fuzzy.html (2523 words)

	Fuzzy Systems - A Tutorial (Site not responding. Last check: 2007-11-05)
		The precision of mathematics owes its success in large part to the efforts of Aristotle and the philosophers who preceded him.
		where m is the membership function, operating in this case on the fuzzy set of old people, which returns a value between 0.0 and 1.0.
		Fuzzy theorists argue that a sting of low membership grades should not produce a high membership grade instead, the limit of the resulting membership grade should be the strongest membership value in the collection.
	www.austinlinks.com /Fuzzy/tutorial.html (2979 words)

	New Page 0 (Site not responding. Last check: 2007-11-05)
		The easiest way to do this is with a membership function based on the person's age.
		Furthermore, membership functions here have been discussed as if they always are based on a single criterion, but this isn't always the case, although it is the most common case.
		One could, for example, want to have the membership function for YOUNG depend on both a person's age and their height (Nikhil's short for his age).
	www.ee.iitb.ac.in /uma/~shreyash/fuzzy (2410 words)

	Bertrand Russell - Free Online Library
		At Trinity College, Cambridge, his brilliance was soon recognized, and brought him a membership in the 'Apostles,’ a forerunner of the Bloomsbury Set.
		It proposed that the foundations of mathematics could be deduced from a few logical ideas.
		In it Russell arrived at the view of Gottlob Frege (1848-1925), that mathematics is a continuation of logic and that its subject-matter is a system of Platonic essences that exist in the realm outside both mind and matter.
	russell.thefreelibrary.com (1670 words)

	Fuzzy Thinking
		We can translate all the theorems about sets into theorems about membership functions, that is, functions satisfying (16).
		Among our generalized membership functions which now include fuzzy membership functions taking values greater than 0 and less than 1, we still have our crisp membership functions which only take the two values 0 and 1.
		They are the same for crisp membership functions, but not for fuzzy ones.
	www.mth.kcl.ac.uk /events/summer_schools/summer_school2004/Landau.html (4206 words)

	Fuzzy Logic
		It is a rule-based AI that tolerates imprecision by using nonspecific terms called membership functions to solve problems.
		Nevertheless, fuzzy logic is a mathematical formalism, and a membership grade is a precise number.
		where m is the membership function, operating in this case on the fuzzy set of old people, which returns a value between 0.0 and 0.1.
	www.cob.ohio-state.edu /~muhanna_1/old/831s95/sart/fuzzy.htm (2157 words)

	DYNAMICAL MODEL TO INVESTIGATE OF FUZZY-NEURO NETWORKS (Site not responding. Last check: 2007-11-05)
		The processes described by neural networks are the dynamic systems and their mathematical model shows that the determination of the weights does not differ from the problem of the solution of the inverse problem which, generally speaking, in the common statement of a question can hardly be investigated.
		The processes described by nets are the dynamic systems and their mathematical model shows that the determination of the weights does not differ from the problem of the solution of the inverse problem which, generally speaking, in the common statement of a question can hardly be investigated.
		As known, an energy function (or Lyapunov function) is a function which is bounded below and is a nonincreasing function of the state of the system [2] For a neural net, the state of the system is the vector of activation of the neurons.
	alaeddin.cc.selcuk.edu.tr /~noval/DynMod.html (1982 words)

	Edward Sazonov: Home Page
		Each linguistic variable has an arbitrary number of associated membership functions: each membership function is represented by a trapezoid.
		Linguistic variables, membership functions, hedges and fuzzy operations are referenced by their respective symbolic names.
		How much a rule's weight will be changed depends both on definition of membership functions for "weight", evaluation result after application of hedges and number of assignments.
	people.clarkson.edu /~esazonov/FuzzyEngine.htm (1447 words)

	Excerpts
		Mathematics has developed powerful tools for studying and dealing with fuzziness - uncertainty, imprecision, vagueness - that researchers encounter when accumulating facts in different fields of their inquiry.
		Fuzziology is not another mathematical study of fuzziness, and social fuzziology is not another field of application of fuzzy logic.
		The problem consists in describing mathematically trajectory of an object, the motion of which is influenced simultaneously by the gravitational forces of three bodies.
	www.zulenet.com /VladimirDimitrov/pages/Excerpts.html (19986 words)

	Fuzzy Set Functions -- from Mathematica Information Center
		Each element of such a list consists of a pair representing the set element and the corresponding value of the membership function.
		Most of the standard set operations and tests for finite fuzzy sets are implemented.
		Two Mathematica sessions (FuzzySet.1 and FuzzySet.2) providing examples which test the functions are included.
	library.wolfram.com /infocenter/MathSource/538 (107 words)

	In: Structures in Mathematical Theories, San Sebastian, 1990, pp (Site not responding. Last check: 2007-11-05)
		Some authors treat a model simply as a mathematical structure, exactly as a set with a system of relations (may be functions) on it (Malzev, 1970; Shoenfield, 1967).
		As mathematical structure M (that is called a model of the language L) the set on which the predicates and functions are defined usually is taken.
		The main progress of contemporary mathematics may be explained as a process of transition from usual (singlenamed) sets to more general cases.
	www.math.ucla.edu /~mburgin/res/math/NSetFouM/NSetFouM.htm (1936 words)

	[No title]
		The coequalizer object k of f:i->j and g:i->j is the subobject of j whose characteristic function p:j->2 is the predicate "is the least representative in its block".
		Core mathematics in the 20th C did not provide a problem for semantics and for that reason the inadequacy of formal set theory was not noticed.
		However, `the' arrow-theoretic account of membership was actually totally omitted from the book, though it should have been in section 4.1, along with the discussion of related basic matters, such as subobjects and their inclusions.
	www.mta.ca /~cat-dist/catlist/1999/set-memb-func-comp (15844 words)

	FUZZY LOGIC (Site not responding. Last check: 2007-11-05)
		Mathematically fuzziness means multivalence so that multivalued fuzziness corresponds to degrees of indeterminacy or ambiguity, partial occurrence of events or relations.
		Membership Function (MF) : A MF is a curve that defines how each point in the universe of discourse is mapped to a value between 0 and 1.
		Membership Functions : This Toolbox includes 11 built-in membership function types, built from several basic functions: piecewise linear functions (triangular and trapezoidal), the Gaussian distribution function (gaussian curves and generalized bell), the sigmoid curve, and quadratic and cubic polynomial curves(Z, S, and Pi curves) :
	www.etse.urv.es /~aoller/fuzzy/fuzzy_logic.htm (2899 words)

	CSISS Classics - Lotfi Zadeh: Fuzzy logic-Incoporating Real-World Vagueness
		The basis of the theory lies in making the membership function lie over a range of real numbers from 0.0 to 1.0.
		However, between 150 and 180 the degree of membership for the class 'tall' can be assigned from the curve varying linearly between 0 and 1.
		In a conventional logic approach, this inherent fuzziness of membership and categorization is not incorporated.
	www.csiss.org /classics/content/68 (1118 words)

	The Process of Learning Mathematics: A Fuzzy set Approach
		The fuzzy sets theory was created in response to the need to have a mathematical representation of such kind of situations.
		But, the knowledge that students have about the various mathematical concepts is usually imperfect, characterized by a different degree of depth.
		This suggests that the application of the fuzzy sets theory in mathematical education could be a very promising tool in order to represent such situations and get useful conclusions.
	www.ipv.pt /millenium/17_ect4.htm (1466 words)

	[No title]
		A function whose image is a bounded set.
		A function of a metric space to itself which moves each point no more than some constant distance.
		] An electron whose wave function is negligible except in the vicinity of an atom.
	www.accessscience.com /Dictionary/B/B27/DictB27.html (2524 words)

	Russell's Paradox [Internet Encyclopedia of Philosophy]
		Russell discovered the contradiction from considering Cantor's power class theorem: the mathematical result that the number of entities in a certain domain is always smaller than the number of subclasses of those entities.
		If there were the same number, there would have to be a 1-1 function f mapping entities in the domain on to subclasses of entities in the domain.
		For example, while it is possible to take the square root of the square root of some number, one cannot simply apply the function square root to the function square root and arrive at a value.
	www.iep.utm.edu /p/par-russ.htm (2709 words)

	A S T R O L O G Y (Site not responding. Last check: 2007-11-05)
		Jyotisa, which is the science of celestial bodies and the eye of the Vedapurusa, consists of 'Skandhas' or sections dealing with mathematics, astronomy and astrology.
		The right day and hour [muhurta] for a function is fixed according to the position of celestial bodies.
		It is not its function to find out how the celestial bodies are beneficial to us or how they may be made favourable to us.
	www.mypurohith.com /Astrology_Page.asp (855 words)

	Nerac spends millions of dollars every year updating and adding science, engineering, patent and business databases. (Site not responding. Last check: 2007-11-05)
		Using the concept of fuzzy mathematics, the authors developed an automated visual inspection system to detect dimple defects of polished wafer surfaces.
		A fuzzy membership function is utilized to cope with the wide range of shape variations of the dimple defects.
		A decision-making mechanism is based on the value of the membership function which describes the pattern's closeness to a dimple.
	nerac.com /site_2/databases/DisplaySearchExample.asp?database=EIX&... (232 words)

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