A fuzzy relation is the cartesian product of mathematical fuzzy sets.[1] Two fuzzy sets are taken as input, the fuzzy relation is then equal to the cross product of the sets which is created by vector multiplication. Usually, a rule base is stored in a matrix notation which allows the fuzzy controller to update its internal values.[2][3]
From a historical perspective, the first fuzzy relation was mentioned in the year 1971 by Lotfi A. Zadeh.[4]
A practical approach to describe a fuzzy relation is based on a 2d table. At first, a table is created which consists of fuzzy values from 0..1. The next step is to apply the if-then-rules to the values. The resulting numbers are stored in the table as an array.
Fuzzy relations can be utilized in fuzzy databases.[5]
^Timothy J. Ross (8 April 2005). Fuzzy Logic with Engineering Applications. John Wiley & Sons. pp. 59–. ISBN 978-0-470-86076-2.
^Galindo, Jos (31 May 2008). Handbook of Research on Fuzzy Information Processing in Databases. IGI Global. pp. 17–. ISBN 978-1-59904-854-3.
^Witold Pedrycz (31 March 1996). Fuzzy Modelling: Paradigms and Practice. Springer Science & Business Media. pp. 39–. ISBN 978-0-7923-9703-8.
^Witold Pedrycz; Fernando Gomide (12 October 2007). Fuzzy Systems Engineering: Toward Human-Centric Computing. John Wiley & Sons. pp. 156–. ISBN 978-0-470-16895-0.
^Zongmin Ma (30 March 2006). Fuzzy Database Modeling with XML. Springer Science & Business Media. p. 97–. ISBN 978-0-387-24249-1.
A fuzzyrelation is the cartesian product of mathematical fuzzy sets. Two fuzzy sets are taken as input, the fuzzyrelation is then equal to the cross...
structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have...
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the...
Fuzzy mathematics is the branch of mathematics including fuzzy set theory and fuzzy logic that deals with partial inclusion of elements in a set on a...
called standard fuzzy set operations; they comprise: fuzzy complements, fuzzy intersections, and fuzzy unions. Let A and B be fuzzy sets that A,B ⊆ U...
fuzzy concepts and fuzzy language is called fuzzy semantics. The inverse of a "fuzzy concept" is a "crisp concept" (i.e. a precise concept). A fuzzy concept...
mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in...
a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain. A binary relation over sets X...
phrased as "a relation on X" or "a (binary) relation over X". An example of a homogeneous relation is the relation of kinship, where the relation is between...
micro-PROLOG [es] of Logic Programming Associates and adds support for fuzzy sets, support logic, and metaprogramming. Fril was originally developed...
relation is called the arity, adicity or degree of the relation. A relation with n "places" is variously called an n-ary relation, an n-adic relation...
The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the...
problems. The Fuzzy VIKOR method has been developed to solve problem in a fuzzy environment where both criteria and weights could be fuzzy sets. The triangular...
entailment: (1) The logical consequence relation relies on the logical form of the sentences: (2) The relation is a priori, i.e., it can be determined...
heap, and enough slight changes will certify the group's heap status – see fuzzy logic. One may object to the first premise by denying that 1,000,000 grains...
function to describe the function that indicates membership in a set. In fuzzy logic and modern many-valued logic, predicates are the characteristic functions...
element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly...
truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible...
membership relation. These include rough set theory and fuzzy set theory, in which the value of an atomic formula embodying the membership relation is not...
the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have...
modelling using fuzzy decision tables". New frontiers in fuzzy logic and computing: 1996 biennial conference of the North American Fuzzy Information Processing...
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