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Operations on fuzzy sets
used operations are called standard fuzzy set operations; they comprise: fuzzy complements, fuzzy intersections, and fuzzy unions. Let A and B be fuzzy sets
Fuzzy_set_operations
Sets whose elements have degrees of membership
In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an
Fuzzy_set
Topics referred to by the same term
sets Set operations (SQL), type of operation in SQL Fuzzy set operations, a generalization of crisp sets for fuzzy sets Set (disambiguation) Set theory
Set_operation
System of logic in computer science
Type-2 fuzzy sets and systems generalize standard type-1 fuzzy sets and systems so that more uncertainty can be handled. From the beginning of fuzzy sets, criticism
Type-2_fuzzy_sets_and_systems
described using fuzzy sets. Rules can connect multiple variables through fuzzy set operations using t-norms and t-conorms. T-norms are used as an AND connector
Fuzzy_rule
Varying application boundaries
represent fuzzy concepts mathematically, using fuzzy logic, fuzzy values, fuzzy variables and fuzzy sets (see also fuzzy set theory). Fuzzy logic is not
Fuzzy_concept
Collection of mathematical objects
§ Basic operations, all elements of sets produced by set operations belong to previously defined sets. In this section, other set operations are considered
Set_(mathematics)
Branch of mathematics
with the introduction of fuzzy sets, the field has since evolved to include fuzzy set theory, fuzzy logic, and various fuzzy analogues of traditional
Fuzzy_mathematics
working fuzzy chip during a technical fair. Defuzzification Fuzzy set Fuzzy set operations "Fuzzy plan with a purpose". Computerworld. Vol. 25, no. 41. 1991-10-14
Fuzzy_electronics
Branch of mathematics that studies sets
mathematics. In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In fuzzy set theory this
Set_theory
Real numbers with a multi-valued logical classification
A fuzzy number is a generalization of a regular real number in the sense that it does not refer to one single value but rather to a connected set of possible
Fuzzy_number
Method to analyze non-binary inputs
A fuzzy control system is a control system based on fuzzy logic – a mathematical system that analyzes analog input values in terms of logical variables
Fuzzy_control_system
Identities and relationships involving sets
performing calculations involving these operations and relations. Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the
Algebra_of_sets
Generalization of the indicator function for classical sets in fuzzy logic
mathematics, the membership function of a fuzzy set is a generalization of the indicator function for classical sets. In fuzzy logic, it represents the degree of
Membership function (mathematics)
Membership_function_(mathematics)
Branch of theoretical mathematics
found throughout the database. Fuzzy concept Fuzzy mathematics Fuzzy set operations Rough set Multiset Category theory Set theory Relational model Burgin
Named_set_theory
Programming language
micro-PROLOG [es] of Logic Programming Associates and adds support for fuzzy sets, support logic, and metaprogramming. Fril was originally developed by
Fril
Fuzzy subalgebras theory is a chapter of fuzzy set theory. It is obtained from an interpretation in a multi-valued logic of axioms usually expressing the
Fuzzy_subalgebra
Standard system of axiomatic set theory
set. Hence the universe of sets under ZFC is not closed under the elementary operations of the algebra of sets. Unlike von Neumann–Bernays–Gödel set theory
Zermelo–Fraenkel_set_theory
Azerbaijani scientist (1921–2017)
fuzzy mathematics, consisting of several fuzzy-related concepts: fuzzy sets, fuzzy logic, fuzzy algorithms, fuzzy semantics, fuzzy languages, fuzzy control
Lotfi_A._Zadeh
Mathematical set that can be enumerated
mathematical set is countable if either it is finite or it can be put in one to one correspondence with the set of natural numbers. Equivalently, a set is countable
Countable_set
French mathematician
the use of aggregation operations in information fusion process" Didier Dubois, Henri Prade (2004) "Interval-valued Fuzzy Sets, Possibility Theory and
Didier_Dubois_(mathematician)
In computer science and operations research, Genetic fuzzy systems are fuzzy systems constructed by using genetic algorithms or genetic programming, which
Genetic_fuzzy_systems
Mathematical set containing no elements
the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories
Empty_set
Mathematical operation with two operands
an element of the set. Examples include the familiar arithmetic operations like addition, subtraction, multiplication, set operations like union, complement
Binary_operation
Paradox in set theory
a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901. Russell's paradox shows that every set theory
Russell's_paradox
Diagram that shows all possible logical relations between a collection of sets
between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships
Venn_diagram
Set that is not a finite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence
Infinite_set
Speculative fiction novel series by Matt Dinniman
MacFarlane's Fuzzy Door Productions, acquired the rights to adapt Dungeon Crawler Carl into a television series. Christopher Yost is set to write the
Dungeon_Crawler_Carl
Finding strings that approximately match a pattern
science, approximate string matching (often colloquially referred to as fuzzy string searching) is the technique of finding strings that match a pattern
Approximate_string_matching
Mathematical set of all subsets of a set
numbers (see Cardinality of the continuum). The power set of a set S, together with the operations of union, intersection and complement, is a σ-algebra
Power_set
Extension of SQL
SQLf is a SQL extended with fuzzy set theory application for expressing flexible (fuzzy) queries to traditional (or ″Regular″) Relational Databases. Among
SQLf
Set whose elements all belong to another set
In mathematics, a set A is a subset of a set B if and only if all elements of A are also elements of B; B is then a superset of A. It is possible for A
Subset
Finite collection of distinct objects
finite sets to be countable.) The free semilattice over a finite set is the set of its non-empty subsets, with the join operation being given by set union
Finite_set
Size of a possibly infinite set
for finite cardinals, these operations coincide with the usual operations for natural numbers. Furthermore, these operations share many properties with
Cardinal_number
Set of the elements not in a given subset
some contexts (for example, Minkowski set operations in functional analysis) it can be interpreted as the set of all elements b − a , {\displaystyle
Complement_(set_theory)
Collection of sets in mathematics that can be defined based on a property of its members
In set theory and its applications throughout mathematics, a class is a collection of mathematical objects (often sets) that can be unambiguously defined
Class_(set_theory)
Logical principle
derived from interviews. Bart Kosko, Fuzzy Thinking: The New Science of Fuzzy Logic, Hyperion, New York, 1993. Fuzzy thinking at its finest but a good introduction
Law_of_excluded_middle
Any one of the distinct objects that make up a set in set theory
mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called A containing the first four
Element_of_a_set
One-to-one correspondence
function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Given
Bijection
Proposition in mathematical logic
specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: There is no set whose
Continuum_hypothesis
Set of elements common to all of some sets
an algebraic operation with sets as operands has been generalized from geometry, where it is encountered in the case of geometric sets of points, such
Intersection_(set_theory)
Logical connective AND
for bitwise operations, where 0 corresponds to false and 1 to true: 0 AND 0 = 0, 0 AND 1 = 0, 1 AND 0 = 0, 1 AND 1 = 1. The operation can also be
Logical_conjunction
Set of all things that may be the input of a mathematical function
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom ( f ) {\displaystyle \operatorname
Domain_of_a_function
Non-contradiction of a theory
\lnot \varphi } are elements of the set of consequences of T {\displaystyle T} . Let A {\displaystyle A} be a set of closed sentences (informally "axioms")
Consistency
Class of mathematical set whose elements are all subsets
In set theory, a branch of mathematics, a set A {\displaystyle A} is called transitive if either of the following equivalent conditions holds: whenever
Transitive_set
Size of a set in mathematics
group cardinality (P1164) (see uses) set cardinality (P2820) (see uses) Cardinal and Ordinal Numbers Fuzzy set § Scalar cardinality Infinitary combinatorics
Cardinality
Set of elements in any of some sets
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through
Union_(set_theory)
Informal set theories
set theory" is a non-formalized theory, that is, a theory that uses natural language to describe sets and operations on sets. Such theory treats sets
Naive_set_theory
Generalization of "n-th" to infinite cases
sets having the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated, although none of these operations are
Ordinal_number
Set with algorithmic membership test
In computability theory, a set of natural numbers is computable (or decidable or recursive) if there is an algorithm that computes the membership of every
Computable_set
Algebraic manipulation of "true" and "false"
they denote the same operation; however, this way of writing Boolean operations allows applying the usual arithmetic operations of integers (this may
Boolean_algebra
Finite sets whose elements are all hereditarily finite sets
mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself
Hereditarily_finite_set
Maximal proper filter
In the mathematical field of set theory, an ultrafilter on a set X {\displaystyle X} is a maximal filter on the set X . {\displaystyle X.} In other words
Ultrafilter_on_a_set
Infinite set that is not countable
mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related
Uncountable_set
Process of repeating items in a self-similar way
scenario that does not use recursion to produce an answer A recursive step — a set of rules that reduces all successive cases toward the base case. For example
Recursion
Alternative to the standard Zermelo–Fraenkel set theory
set theory Morse–Kelley set theory Tarski–Grothendieck set theory Ackermann set theory Type theory New Foundations Positive set theory Internal set theory
List of alternative set theories
List_of_alternative_set_theories
Axiom of set theory
an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, one can identify another set containing one
Axiom_of_choice
Mathematical set containing all objects
In set theory, a universal set is a set that contains all of the objects in the theory, including itself. In set theory as usually formulated, it can
Universal_set
System in mathematical set theory
In mathematics, vague sets are an extension of fuzzy sets. In a fuzzy set, each object is assigned a single value in the interval [0,1] reflecting its
Vague_set
Mathematical logic concept
In computability theory, a set S of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable
Computably_enumerable_set
Approximate string matching algorithm
information on these operations. The implementation below performs fuzzy matching (returning the first match with up to k errors) using the fuzzy bitap algorithm
Bitap_algorithm
Elements in exactly one of two sets
of sets Boolean function Complement (set theory) Difference (set theory) Exclusive or Fuzzy set Intersection (set theory) Jaccard index List of set identities
Symmetric_difference
Finite ordered list of elements
n-tuple can be formally defined as the image of a function that has the set of the first n natural numbers as its domain (1, 2, ..., n). Tuples may be
Tuple
Operations research that evaluates multiple conflicting criteria in decision making
and Cooper, 1961). Fuzzy-set theorists Fuzzy sets were introduced by Zadeh (1965) as an extension of the classical notion of sets. This idea is used in
Multiple-criteria decision analysis
Multiple-criteria_decision_analysis
Decision-making strategy
alternative to the ideal solution. The fuzzy operations and procedures for ranking fuzzy numbers are used in developing the fuzzy VIKOR algorithm. Rank reversals
VIKOR_method
Basic framework of mathematics
generality of algebra, which consisted to apply properties of algebraic operations to infinite sequences without proper proofs. In his Cours d'Analyse (1821)
Foundations_of_mathematics
3-volume treatise on mathematics, 1910–1913
set is the starting set, and other symbols can appear but only by definition from these beginning symbols. A starting set might be the following set derived
Principia_Mathematica
Pair of logical equivalences
complement of the union of two sets is the same as the intersection of their complements The complement of the intersection of two sets is the same as the union
De_Morgan's_laws
Concept in mathematical logic
In set theory, a hereditary set (or pure set) is a set whose elements are all hereditary sets. That is, all elements of the set are themselves sets, as
Hereditary_set
applied fuzzy logic and fuzzy set theory as a theoretical basis for approximate reasoning. T-norm fuzzy logics belong in broader classes of fuzzy logics
T-norm_fuzzy_logics
Set theory concept
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary
Von_Neumann_universe
Use of braces for specifying sets
{Z} ,n=2k\}} — The set of all even integers, expressed in set-builder notation. In mathematics and more specifically in set theory, set-builder notation
Set-builder_notation
Branch of mathematics
∧ can be viewed as a total binary operation in the sense of universal algebra. Hence, in a lattice, two operations ∧ and ∨ are available, and one can
Order_theory
Particular class of sets which can be described entirely in terms of simpler sets
in set theory, the constructible universe (or Gödel's constructible universe), denoted by L , {\displaystyle L,} is a particular class of sets that
Constructible_universe
mathematical objects include numbers, expressions, shapes, functions, and sets. Mathematical objects can be very complex; for example, theorems, proofs
Mathematical_object
Axioms for the natural numbers
multiplication operations are directly included in the signature of Peano arithmetic, and axioms are included that relate the three operations to each other
Peano_axioms
contradictions within modern axiomatic set theory. Set theory as conceived by Georg Cantor assumes the existence of infinite sets. As this assumption cannot be
Paradoxes_of_set_theory
Mathematical set formed from two given sets
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is an
Cartesian_product
Branch of mathematical logic
foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of
Reverse_mathematics
System of mathematical set theory
Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory
Zermelo_set_theory
Logic with discrete truth values
Weisstein, Eric (2018). "Fuzzy Logic". MathWorld--A Wolfram Web Resource. Klawltter, Warren A. (1976). Boolean values for fuzzy sets. Theses and Dissertations
Finite-valued_logic
Concept in axiomatic set theory
power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. It guarantees for every set x {\displaystyle x} the existence of a set P ( x
Axiom_of_power_set
Method of deriving conclusions
"Meditative Fuzzy Logic: A New Approach for Contradictory Knowledge Management". In Nikravesh, Masoud; Zadeh, Lofti A. (eds.). Forging New Frontiers: Fuzzy Pioneers
Rule_of_inference
Subfield of mathematics
logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic. Lindström's theorem implies that the only extension of first-order
Mathematical_logic
Theory that allows sets to be elements of themselves
ISBN 978-1-139-47927-1. Nicolás Sevilla Simón (2025). "On the consistency of 𝑁𝐹 via Fuzzy Forcing". arXiv:2504.14400 [math.LO]. Pakkan & Akman (1994), section link
Non-well-founded_set_theory
Symbol representing a mathematical object
often numbers. More specifically, the values involved may form a set, such as the set of real numbers. The object may not always exist, or it might be
Variable_(mathematics)
Mathematical concept
∼ . {\displaystyle S/{\sim }.} When the set S {\displaystyle S} has some structure (such as a group operation or a topology) and the equivalence relation
Equivalence_class
System of mathematical set theory
of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of
Morse–Kelley_set_theory
discussed below are provably independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Infinite cardinal number
respect to finite operations—sums, products, etc. The process involves defining, for each countable ordinal, via transfinite induction, a set by "throwing
Aleph_number
Mathematician (1845–1918)
January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established
Georg_Cantor
Concept in axiomatic set theory
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation (Aussonderungsaxiom)
Axiom_schema_of_specification
Form of mathematical proof
elements of any well-founded set, that is, a set with an irreflexive relation < that contains no infinite descending chains. Every set representing an ordinal
Mathematical_induction
Statement that is taken to be true
Zermelo–Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann–Bernays–Gödel set theory, a conservative
Axiom
Theorem for proving more complex theorems
product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal
Lemma_(mathematics)
Symbolic description of a mathematical object
the domain. (Usually letters like x, or y) A set of operations: Function symbols representing operations that can be performed on elements over the domain
Expression_(mathematics)
Problem in computer science
"non-trivial" means that the set of partial functions that satisfy the property is neither the empty set nor the set of all partial functions. For example
Halting_problem
Set with exactly one element
a singleton (also known as a unit set or one-point set) is a set with exactly one element. For example, the set { 0 } {\displaystyle \{0\}} is a singleton
Singleton_(mathematics)
Axiom of Zermelo-Fraenkel set theory
define ∅ {\displaystyle \varnothing } to be the empty set and recognize the successor operation: ∃ I ( ∅ ∈ I ∧ ∀ x ( x ∈ I ⇒ ( x ∪ { x } ) ∈ I ) ) . {\displaystyle
Axiom_of_infinity
Equalities for combinations of sets
expressions, and performing calculations, involving these operations and relations. The binary operations of set union ( ∪ {\displaystyle \cup } ) and intersection
List of set identities and relations
List_of_set_identities_and_relations
FUZZY SET-OPERATIONS
FUZZY SET-OPERATIONS
Boy/Male
Egyptian Hebrew Swedish
Son of Seb and Nut.
Surname or Lastname
English and German
English and German : topographic name for someone who lived by the sea-shore or beside a lake, from Middle English see ‘sea’, ‘lake’ (Old English sǣ), Middle High German sē. Alternatively, the English name may denote someone who lived by a watercourse, from an Old English sēoh ‘watercourse’, ‘drain’.
Surname or Lastname
English
English : perhaps a variant of Sait, from the Old English personal name Sǣgēat (‘sea Geat’).
Female
Egyptian
, the wife of Osirtesen.
Female
Egyptian
, a sister of Sekherta.
Female
Egyptian
, the mother of Fai-hor-ou-oer.
Female
Egyptian
, the wife of the usurper Sipthah.
Male
English
Anglicized form of Hebrew Sheth, SETH means "buttocks." In the bible, this is the name of the third son of Adam and Eve. Compare with other forms of Seth.
Female
Hungarian
Hungarian form of Greek Elisabet, ERZSÉBET means "God is my oath."
Female
English
Short form of English Elizabeth, BET means "God is my oath."Â
Male
Egyptian
, the seven great spirits of the Ritual of the Dead.
Male
Hebrew
Variant spelling of Hebrew Sheth, SHET means "buttocks."
Female
Egyptian
, an uncertain goddess.
Female
Egyptian
, second wife of Antef.
Male
Egyptian
, the seven great spirits of the Ritual of the Dead.
Male
Hindi/Indian
(सेठ) Hindi name derived from the Sanskrit word setu, SETH means "bridge." Compare with other forms of Seth.
Surname or Lastname
English
English : variant spelling of See.
Female
Egyptian
, a wife and daughter of Antef.
Female
Egyptian
, a sister of Sekherta.
Male
English
Short form of English Stephen, STE means "crown."
FUZZY SET-OPERATIONS
FUZZY SET-OPERATIONS
Boy/Male
Tamil
Prahlad | பà¯à®°à®¹à®²à®¾à®¤
Excess of Joy
Male
English
Anglicized form of Greek Hananias, ANANIAS means "whom Jehovah has graciously given." In the New Testament bible, this is the name of the husband of Sapphira, a Christian at Damascus, and a son of Nedebaeus (Greek Nabadias).
Girl/Female
Indian, Punjabi, Sikh
Comfort; Repose; Strong Person
Male
Italian
Short form of Italian Giovanni, VANNI means "God is gracious."
Girl/Female
Bengali, Hindu, Indian, Kannada, Marathi, Sanskrit, Sindhi, Telugu
A River
Girl/Female
Arabic
Precious
Boy/Male
Arabic, Bengali, Indian, Muslim
Granted; Blessed; Normal Man; Men with All Blessings of God
Boy/Male
Hindu, Indian, Punjabi, Sikh
Imbued in the Lord's Absorption
Surname or Lastname
English
English : variant of Roderick.
Boy/Male
American, Australian, British, Chinese, Christian, English, German
Baker
FUZZY SET-OPERATIONS
FUZZY SET-OPERATIONS
FUZZY SET-OPERATIONS
FUZZY SET-OPERATIONS
FUZZY SET-OPERATIONS
n.
Furnished with fuzz; having fuzz; like fuzz; as, the fuzzy skin of a peach.
v. i.
To fit or suit one; to sit; as, the coat sets well.
a.
Fixed in position; immovable; rigid; as, a set line; a set countenance.
n.
Direction or course; as, the set of the wind, or of a current.
v. t.
To determine; to appoint; to assign; to fix; as, to set a time for a meeting; to set a price on a horse.
v. t.
To cause to sit; to make to assume a specified position or attitude; to give site or place to; to place; to put; to fix; as, to set a house on a stone foundation; to set a book on a shelf; to set a dish on a table; to set a chest or trunk on its bottom or on end.
a.
Firm; unchanging; obstinate; as, set opinions or prejudices.
n.
See Set, n., 2 (e) and 3.
a.
Regular; uniform; formal; as, a set discourse; a set battle.
v. t.
To compose; to arrange in words, lines, etc.; as, to set type; to set a page.
n.
The state or quality of being muzzy.
v. i.
To be fixed for growth; to strike root; to begin to germinate or form; as, cuttings set well; the fruit has set well (i. e., not blasted in the blossom).
v. t.
To put in order in a particular manner; to prepare; as, to set (that is, to hone) a razor; to set a saw.
n.
Not firmly woven; that ravels.
n.
That which is set, placed, or fixed.
v. t.
To establish as a rule; to furnish; to prescribe; to assign; as, to set an example; to set lessons to be learned.
a.
Established; prescribed; as, set forms of prayer.
a.
Furzy; gorsy.
n.
A young plant for growth; as, a set of white thorn.
imp. & p. p.
of Set