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math-debrief
  • Math Debrief
  • Math: TIMELINE
  • 100-fundamentals
    • debrief-name: math section-code: 000 section-name: general section-desc: Elementary topics pervasive
      • About Mathematics
      • abstraction-in-math
      • About Math
      • Axiom schema
      • Basic concepts in math
      • Collections
      • Elementary concepts in objects
      • Elements of mathematics
      • math-as-a-language
      • Mathematical structures
      • List of mathematics-based methods
      • Mathematics and Reality
      • Mathematics: General
      • Controversial mathematics
      • the-elements-of-math
      • What is mathematics
    • The foundation of mathematics
      • Mathematical foundations
      • Foundations of Mathematics
      • Axiomatization of mathematics
      • Foundational crisis of mathematics
      • Foundations
      • Hilbert's problems
      • impl-of-math-in-set-theory
      • GΓΆdel's Incompleteness Theorem
      • Theorems in the foundations of mathematics
      • The list of FOM candidates
      • Logicism
    • Philosophy of mathematics
      • Constructive mathematics
      • Constructive mathematics
      • Metamathematics
      • Philosophy of mathematics
      • Schools of mathematics
    • terms
      • Terms
      • Arithmetic
      • Axiom
      • The Axiomatic Method
      • discrete-math
      • 201 Discrete mathematics
      • Euclidean space
      • Formal system
      • Function
      • Generalization
      • Geometry
      • Higher-order
      • Impredicativity
      • Level of measurement
      • Mathematical definition
      • FAQ
      • Mathematical function
      • Mathematical induction
      • Mathematical object
      • Mathematical object
      • Equivalent definitions of mathematical structures
      • Mathematics
      • Mathematical model
      • mathematical-notation
      • Mathematical pages
      • Mathematical terminology
      • Mathematical adjective
      • Numbers
      • plane
      • Primer: Set Theory
      • Mathematical primitive
      • Set
      • Space
      • theory
      • Variable
  • 200 Set and Set theory
    • Sets: Hierarchy
    • set.TERMS
    • SETS β€Ί TOPICS
    • 201 Set concepts
      • Mathematical collections
      • The notion of sets
      • Specification of sets
    • Set cardinality
      • Cardinality of the continuum
      • Cardinality
      • Set Cardinality
      • cardinality2
      • Set cardinality
    • Set operations
      • Disjoint sets
      • Overlapping sets
      • Product
      • set-interactions
      • Set qualities
      • Set relations
    • Set properties
      • Basic set properties
      • Set properties
    • Set relations
      • Basic set relations
      • Disjoint sets
      • Inclusion relation
      • Membership Relation
      • Set membership
    • Summary
      • Set FAQ
      • Sets: Summary
    • Set theories
      • Axiomatic set theory
      • Set Theories
      • Naive Set Theory
      • Morse-Kelley set theory
      • von Neumann-Bernays-GΓΆdel Set Theory
      • Quine's New Foundations (NF)
      • Cantor's set theory
      • Zermelo-Fraenkel set theory
    • Axioms of set theory
      • axiom-of-choice
      • The Axiom of Extensionality
      • Axiom of infinity
      • axiom-of-pairing
      • Axiom of powerset
      • Axiom of Regularity
      • Axiom of replacement
      • Axiom of union
      • Axiom of well-ordering
      • axiom-schema-of-comprehension
      • Axiom Schema of Specification
      • Axioms of set theories
      • List of axioms in set theory
      • ZFC Axioms
    • Sets: Terms
      • Bell Number
      • Cardinal number
      • Class (set theory)
      • Closure
      • empty-set
      • Extended set operations
      • Extensions by definitions
      • Family of sets
      • Fundamental sets
      • fundamental-sets2
      • Georg Cantor
      • History of set theory
      • Implementation of mathematics in set theory
      • Indexed family of sets
      • Extensional and intensional definitions
      • Involution
      • list-of-axioms-of-set-theory
      • Implementation of mathematics in set theory
      • Set membership
      • Naive Set Theory
      • Number of relations
      • empty-relation
      • Set Partitioning
      • Powerset
      • Russell's paradox
      • Set-builder notation
      • Set equivalence
      • Set Notation in latex
      • Set notation
      • Set partition
      • Intensional and extensional set specification
      • Set notation
      • Basic concepts in set theory
      • set-theory
      • Set Types
      • set
      • subset
      • Transfinite number
      • Tuples
      • ur-elements
  • Relations
    • basic-concepts
      • algebraic-axioms
      • Elements of a relation
      • Types of Relations
      • Named Relations
      • Relation theory
      • Relations
      • Types of relations
    • Relations
      • Definitions
      • Reflexivity
      • Symmetry
      • Transitivity
    • relation-properties
      • Uniqueness properties of relations
    • Types of relations
      • Transitivity
      • Binary Relation
      • Congruence relation
      • Connex relation
      • axioms-sets-zfc
      • Endorelation
      • Equivalence relation
      • Euclidean
      • Finitary relation
      • Heterogeneous relation
      • Homogeneous relation
      • Transitivity
      • Partial equivalence relation
      • Transitivity
      • Transitivity
      • Reflexive relation
      • Reflexivity
      • Index of relations
      • Serial relation
      • Symmetry
      • Transitivity
      • Ternary relation
      • Trichotomy
      • Universal relation
      • Well-foundedness
    • terms
      • Relations
      • Binary relation
      • Relations
      • _finitary-rel
      • Relations: Overview
      • Relations
      • Index of relations
      • Binary relations
      • Composition of relations
      • Equivalence class
      • Notation
      • Relation
      • Relations
      • Sets: Summary
      • Aggregation: Sets, Relations, Functions
  • Order theory
    • Order theory
    • List of order structures in mathematics
    • List of order theory topics
    • Order theory
      • Hasse diagram
      • Order theory
      • ordered-set
      • Partial order
      • Partially ordered set
      • Total order
  • Function Theory
    • Function Theory: GLOSSARY
    • Function Theory: HIERARCHY
    • Function Theory: LINKS
    • Function Theory: TERMS
    • Function Theory: TOPIC
    • Function Theory: WIKI
    • _articles
      • about-functions
      • Function
      • Formal definition
      • Definition
      • constant
      • Introduction
      • Types of functions
      • Functions: Summary of Notations
      • Functions: Overview
      • Properties of functions
      • Function properties
      • Functions: Summary
      • Function
    • Abjections
      • Bijective function
      • Function (abjections)
      • Injective function
      • Surjective function
    • topics
      • Function: TERMS
      • Codomain
      • Composition of functions
      • Currying
      • Division of functions
      • Domain
      • Function fixed points
      • Function cardinality
      • Function definition
      • Elements of a function
      • Function in mathematics
      • Function notion
      • Function operations
      • Function properties
      • Functional statements
      • Functions in programing languages
      • Image and Preimage
      • Image
      • Inverse function
      • Notion of functions
      • Number and types of functions between two sets
      • Operation
      • Range
      • Successor function
      • Time complexity classes
  • debrief-name: math section-code: 280 section-name: domain-theory section-desc:
    • Domain theory: LINKS
    • Domain theory
  • Logic
    • Logic: CHRONO TERMS
    • Logic: CLUSTERS
    • lo.GLOSSARY
    • Logic: Wiki links
    • 305-basic-concepts
      • Introduction to Logic
      • Argumentation
      • Logic: Basic terminology
      • Logic: Terminology
      • Truth function
      • Truth function
    • README
      • Mathematical Logic
      • Types of Logic
      • BHK interpretation
      • FOL
      • Index of Logic Forms
      • History of logic
      • Logic Indices
      • Interpretation of symbols in logic and math
      • logic-systems
      • Mathematical Logic: People and Events
      • Index of logical fallacies
      • Logical symbols
      • Mathematical conjecture
      • Mathematical induction
      • Mathematical lemma
      • Mathematical Logic
      • Mathematical proof
      • Mathematical theorem
      • Mathematical theory
      • Monotonicity of entailment
      • Satisfiability Modulo Theories
      • Sequent Calculus
      • Sequent
      • Tableaux
      • Truth tables
    • 360-propositional-logic
      • Propositional Logic
      • Propositional Logic
    • 370-predicate-logic
      • Predicate Logic
      • First-order logic
      • Predicate calculus
      • Examples of predicate formulae
    • 380-proof-theory
      • Argument-deduction-proof distinctions
      • Direct proof
      • Mathematical induction
      • Mathematical induction
      • Mathematical proof
      • Natural deduction
      • Natural deduction
      • Proof by induction
      • Proof by induction
      • proof-calculus
      • Proof Theory
      • Structural induction
      • System L
      • Proof theory
    • Logic: Indices
      • GΓΆdel's Incompleteness Theorem
      • The History of Mathematical Logic
      • forallx
      • Logic for CS
      • Lectures in Logic and Set Theory
      • _logicomix
    • Logic
      • Logical connectives
      • Logical equivalence
    • Rules of Inference
      • WIKI
      • Conjunction elimination
      • Conjunction introduction
      • Cut rule
      • Disjunction elimination
      • Disjunction introduction
      • Disjunctive syllogism
      • Exportation
      • implication-elimination
      • implication-introduction
      • Rules of Inference: Index
      • Rules of inference
      • Rules of Inference for Natural Deduction
      • Logical Inference
      • Reiteration
      • Rule of inference
      • Structural rules
      • substitution
    • Logic
      • The principle of bivalence
      • The principle of explosion
      • The Law of Identity (ID)
      • Laws of thought
      • Properties of logic systems
      • List of laws in logic
      • The law of non-contradiction
    • Logic
      • Logic systems: LINKS
      • Logic system
      • logic-systems
      • logic-typ
      • logics-by-purpose
      • _logics
      • Affine logic
      • Algebraic logic
      • Bunched logic
      • Classical logic
      • Traditional first-order logic
      • Hoare logic
      • Linear logic
      • Modal logic
      • Non-monotonic logic
      • Syntax
      • Predicate logic
      • Propositional Logic
      • Relevance logic
      • Separation logic
      • Substructural logics
      • Syllogistic logic
    • Logic: Sections: Elementary
    • Logic: Topics
      • Pages in Logic
      • Logic ❱ Terms ❱ List
      • Logic ❱ Terms ❱ Definitions
      • Absoluteness
      • Assumption
      • Automated theorem proving
      • Canonical normal form
      • Categorical proposition
      • Classical linear logic
      • Consequence
      • Decidability
      • Deduction systems
      • deduction-theorem
      • Deductive reasoning
      • Diagonal lemma
      • Fallacy
      • Fitch notation
      • Formal language
      • formal-system
      • Formalism
      • Formula
      • functionally complete
      • Hilbert system
      • Hoare logic
      • horn-clause
      • Mathematical induction
      • Induction
      • Inductive Reasoning
      • Intuitionistic logic
      • Intuitionistic logic
      • Intuitionistic logic
      • Judgement
      • Judgments
      • Linear logic
      • Logic in computer science
      • Logic
      • Logical connective
      • Logical consequence
      • Logical constant
      • Logical form
      • axioms-sets
      • Logical reasoning
      • Ludics
      • Non-logical symbol
      • Predicate
      • Premise
      • Quantification
      • Realizability
      • Boolean satisfiability problem
      • DPLL algorithm
      • Satisfiability
      • Semantics of logic
      • Skolemization
      • SAT and SMT
      • Syntax
      • Tautology
      • Term
      • Unification
      • Validity
  • 510 Lambda Calculi
    • Lambda Calculus: GLOSSARY
    • Lambda calculi: LINKS
    • Lambda Calculus: OUTLINE
    • Lambda Calculus: Basic concepts
      • Introduction
      • Lambda expressions
      • Free variables
    • Lambda Calculi
      • Lambda calculus: LINKS
      • Lambda calculus combinators in Haskell
      • Lambda calculus: Combinators
      • Combinators
      • combos-all.js
      • combos-bird.js
      • combos-birds-list.js
      • combos-birds.js
      • Fixed-point combinator
      • Fixpoint operator
      • Lambda calculus: Fixpoint
    • combinatory-logic
      • algebraic-structures
      • Combinatory logic
      • Combinatory logic
      • relation-classification
      • 04-definition
    • Lambda calculus encoding schemes
      • bohm-berarducci-encoding
      • Index of Church encodings
      • Church encodings
      • Church Numerals
      • Encoding data structures
      • Encoding schemes in lambda calculi
      • Lambda encoding
      • Mogensen-Scott encoding
      • Parigot encoding
      • encodings
        • Encoding data structures
        • Encoding of Data Types in the Ξ»-calculus
        • church-booleans
        • Church data structures
        • Church encoding
        • Church Numerals: Church encoding of natural numbers
        • Lambda Calculus: Church encoding
        • Lambda Calculus: Church encoding
        • church-numerals
        • Lambda Calculus: Church encoding: Numerals
        • Church pair
        • Pair
        • Lambda Calculus: Church encoding
        • Alternative encodings
        • Encoding schemes
        • Encoding schemes
        • Encodings in Untyped Lambda Calculus
        • Lambda calculus
        • Scott encoding
        • Lambda calculus: Scott encoding
    • lambda-calculus-evaluation
      • Call-by-name
      • Call-by-need
      • Call-by-value
    • lambda-calculus-forms
      • Beta normal form
      • Lambda terms
      • Fixity of lambda-terms
    • lambda-calculus-reductions
      • Alpha conversion
      • Beta reduction
      • Delta reduction
      • Eta conversion
      • Eta conversion
      • Lambda calculus: Ξ·-conversion
    • lambda-calculus
      • Alonzo Church
      • Inference rules for lambda calculus
      • Lambda Calculus: Introduction
      • Lambda abstraction
      • Lambda application
      • Lambda Calculus: Definition
      • About Ξ»-calculus
      • Type inference
      • Lambda Calculus
      • Lambda Calculus: Introduction
      • Introduction to Ξ»-calculus
      • Lambda calculus
      • Definition of Lambda Calculus
      • Functions in lambda calculus
      • History of Lambda Calculus
      • Using the Lambda Calculus
      • Name capturing
      • Variable occurrences
      • Variables
    • Lambda Calculus
      • Church-Rosser theorem
      • Curry's paradox
      • De Bruijn index
      • de Bruijn notation
      • Deductive lambda calculus
      • Kleene-Rosser paradox
      • Aspects of the lambda calculus
      • Function Refactoring
      • Lambda lifting
      • Let expression
      • Reduction strategy
      • Substitution
    • typed-lambda-calculi
      • Lambda Cube
      • Simply typed lambda calculus
      • System F
      • Typed lambda calculi
  • Type theory
    • Type Theory: GLOSSARY
    • Type theorists
    • Type Theory: SUMMARY
    • TERMS: Type Theory
      • Types
      • History of type theory
      • History of Type Theory
    • curry-howard-correspondence
      • The Curry-Howard Correspondence in Haskell
      • Curry-Howard correspondence
      • Curry-Howard correspondence
      • Curry-Howard correspondence
      • Curry-Howard-Lambek correspondence - HaskellWiki
    • dependent-types
      • Dependent type
      • Dependent type
    • Hindley-Milner type-system
      • Hindley-Milner type system
      • Monomorphism vs polymorphism
      • Let-polymorphism
      • The Hindley-Milner type system
      • Algorithm W in Haskell
      • Hindley-Milner Type Inference: W Algorithm
      • hindley–milner-type-system
      • Hindley-Milner type system
      • HM inference examples
      • HM in ML
      • Type Inference
    • Homotopy type theory
      • Homotopy type theory
      • Univalent Type theory as the foundations of mathematics
    • Intuitionistic type theory
      • Inductive definition
      • Inductive type
      • Intuitionistic type theory
    • Type Theory
      • TTTools
      • Coinduction
      • Impredicativity
      • Lean
      • Subsumption
    • Type Theory : Topics
      • Type Theory : Terms
      • Recursion types
      • Recursive data type
      • Subtyping
      • Type Class
      • Type Equivalence
      • Type Inference
      • Type rule
      • Type system
      • Variance
    • type-theories
      • Calculus of Constructions
      • Constructive type theory
      • ramified-type-theory
      • simple-type-theory
      • Substructural type systems
    • type-theory-general
      • Linear types
      • History of Type Theory
      • Type Theory
      • Overview
      • Type Theory
  • Abstract Algebra
    • 410-group-theory
      • Abelian group
    • algebras
      • Associative Algebra
      • Field
      • Group-like algebraic structures
      • group
      • Lattice
      • Magma
      • monoid
      • Overview of Algebras
      • Quasigroup
      • Rack and quandle
      • Ring
      • Semigroup
      • Algebra of sets
      • Setoid
    • boolean-algebra
      • Boolean algebra
      • Axioms in Boolean Algebra
      • Boolean algebra
      • Boolean Algebra Laws
      • Boolean Algebra Laws
      • Two-element Boolean algebra
      • Boolean algebra
      • Boolean domain
    • terms
      • Algebra
      • Axioms of abstract algebra
      • Algebraic notation for algebraic data types
      • Algebraic structure
      • Algebraic structure
      • Field of sets
      • Homomorphism
      • Isomorphism
      • Algebraic structures
      • Mathematical structure
      • Polynomials
      • Relation algebra
  • Category Theory
    • CT GLOSSARY
    • Category Theory: OUTLINE
    • CT SUMMARY
    • A First Introduction to Categories (2009)
      • Sets, maps, composition
      • 02-history
      • axioms-logic
      • Bijection of functions
      • Commutative diagram
      • Directed graph
      • CT prerequisites
      • String diagram
      • Transitive closure
    • Category Theory Fundamentals
      • Introduction
      • Interpretation
      • Fundamental concepts
      • Category theory
      • Category
      • Category Theory: Definitions
    • Key concepts
      • Duality
      • Functor
      • Homeset
      • Initial Object
      • Morphism
      • Natural transformation
      • Object
      • Terminal Object
    • Categorical constructions
      • Categorical constructions
      • Coproduct
      • Diagram
      • Product
      • Universal construction
    • Types of categories
      • Concrete category
      • Discrete category
      • Functor category
      • Groupoid
      • Hask
      • Kleisli category
      • Locally small category
      • Monoid
      • monoidal-categories.md
      • Index of named categories
      • Opposite category
      • Ordered category
      • Set category
      • Small category
      • Subcategory
    • Types of Functors
      • Adjoint functor
      • relation-arity
      • Endofunctor
      • Faithful functor
      • Forgetful functor
      • Hom functor
      • Identity functor
      • Inverse functor
      • Monad
      • Powerset functor
    • Types of Morphisms
      • Anamorphism
      • Automorphism
      • Catamorphism
      • Endomorphism
      • Epimorphism
      • Homomorphism
      • Hylomorphism
      • Idempotent morphism
      • Identity morphism
      • Inverse morphism
      • Isomorphism
      • Metamorphism
      • monomorphism
      • Natural isomorphism
      • Split morphism
    • 20-advanced-concepts
      • Coalgebra
      • (Co)Inductive types
      • Recursion Schemes
    • Category Theory
      • Category Theory: TERMS
      • Algebraic Data Types
      • Category Theory
      • Category
      • Coproduct
      • Function type
      • Functoriality
      • Initial Object
      • Limits and Colimits
      • Natural Transformation
      • 5. Products
      • Terminal Object
    • Category Theory :: Contents
      • CT :: Links
      • Category Theory :: Terms
      • Category :: Definition
      • F-Algebra
      • Functor
      • Initial object
      • Monoid
      • Natural Transformation
      • Number of morphisms
      • Terminal object
      • Transitive closure
      • Types of morphisms
      • Categories by cardinality
      • Types of functors
  • Number Theory
    • Invariance and Monovariance Principle
    • 615-arithmetic
      • Addition
      • Aliquot sum
      • Arithmetic function
      • Laws
      • Arithmetic operations
      • Index of arithmetic operations
      • Arithmetic operations
      • Arithmetic
      • Divisibility rules
      • Divisibility
      • division
      • Divisor Function
      • Divisor Summatory Function
      • Divisor
      • Euclidean division
      • Hyperoperations
      • hyperops
      • Modular arithmetic
      • Multiplication
      • Number Theory: primer in numbers
      • Percentage
      • Rules of Divisibility
      • Subtraction
    • The fundamental sets of numbers
      • Algebraic numbers
      • Complex numbers
      • Fractions
      • Fundamental number sets
      • Imaginary numbers
      • Integers
      • Irrational numbers
      • Natural number
      • Rational numbers
      • Real numbers
      • Transcendental numbers
      • Ulam's spiral
      • The whole numbers
    • COUNTING THEORY
      • Counting Theory
      • counting
      • Fundamental Counting Rules
    • 630-combinatorics
      • Combinatorics
      • Combinations
      • Combinatorics
      • Counting theory
      • Counting theory
      • Enumerative combinatorics
      • Partition
      • Pascals triangle
      • Permutations
      • Twelvefold way
    • Probability theory
      • Statistics β€Ί Probability theory: Glossary
      • Statistics β€Ί Probability theory β€Ί Topics
      • Statistics β€Ί Probability theory β€Ί Wiki Links
      • Conditional Probability
      • Distribution
      • Probability theory
      • Probability
    • Number theory
      • euclids-lemma
      • gcd-lcm
      • Induction
      • Infinity
      • Numbers and numerals with interesting properties
      • Lagrange's four-square theorem
      • Matrix
      • Matrix
      • List of Number Systems
      • Number Theory
      • Number Theory with Glenn Olsen
      • Number
      • Arithmetic
      • Numbers
      • numeral-prefixes
      • Numeral system
      • Numeral
      • Ordinal numbers
      • Parity
      • Peano axioms
      • Polynomial
      • Polynomial
      • Positional notation
      • Probability
      • Symbol
      • Well Ordering Principle
    • topics
      • Coprimality
      • Facorization of composite numbers
      • Fundamental Theorem of Arithmetic
      • Prime factorisation
      • Prime number
      • Prime numbers
  • Theory of computation
    • Theory of computation: Abbreviations
    • Theory of computation: CHRONOLOGICAL TOPICS
    • Theory of computation: GLOSSARY
    • Theory of Computation: HIERARCHY
    • Theory of computation: LINKS
    • Theory of computation: TERMS
    • Theory of computation: TOPICS
    • Theory of computation: WIKI
    • Theory of Computation
      • _toc-more
      • Theory of Computation
    • 610-automata-theory
      • Abstract machine
      • Automata Theory
      • Automaton
      • Edit distance
      • Finite-state Machine
      • Automata Theory: WIKI
    • Formal systems
      • Abstract interpretation
      • Alphabet
      • Binary combinatory logic
      • Chomsky hierarchy
      • Epsilon calculus
      • Formal language
      • Iota and Jot
      • Regular expression
      • Regular Language
      • SKI combinator calculus
    • 621-grammar
      • Backus-Naur Form (BNF)
      • Context-free grammar
      • Context-sensitive grammar
      • Extended Backus–Naur Form (EBNF)
      • Regular Language
      • Terminal and nonterminal symbols
    • 622-syntax
      • Syntax
    • 624-semantics
      • Axiomatic semantics
      • Denotational Semantics: Summary
      • Denotational Semantics
      • Denotational Semantics
      • Denotational semantics
      • Formal semantics
      • Operational semantics
      • Semantics in CS
      • Semantics
    • 630-computability-theory
      • Computability (recursion) theory: TERMS
      • Computability (recursion) theory: TOPICS
      • Effective Computability
      • Church Thesis
      • Church-Turing Thesis
      • Computability theory
      • Computability
      • Computable function
      • Entscheidungsproblem
      • Halting problem
      • Machine that always halts
      • McCarthy Formalism
      • Super-recursive algorithm
      • Recursion theory
    • 632-recursive-function-theory
      • Recursion Theory
      • Ackermann function
      • General recursive function
      • Minimization operator
      • Partial functions
      • Recursion Function Theory
      • Sudan function
    • 634-primitive-recursive-functions
      • Primitive Recursive Function
      • Initial functions
      • The list of primitive recursive functions
      • Primitive combination
      • Primitive composition
      • Primitive recursion
      • Successor function
    • 640-models-of-computation
      • Models of computation: Summaries
      • Model of computation
    • 680-complexity-theory
      • Algorithmic Complexity
      • Complexity Theory
  • debrief-name: math section-code: 900 section-name: aggregations section-desc: Aggregations, indices,
    • Index of closures
    • List of mathematical entities
    • List of mathematical objects
    • Enumeration of mathematical structures
    • Math : Axioms as Formulae
    • 950-math-areas
      • Areas of mathematics
      • Areas of mathematics
    • 970-links
      • check
      • Math: Links
      • Math Debrief: Links
      • Math Primer: LINKS
      • Links
      • Math: LINKS: ncatlab
      • Math: LINKS
      • WIKI
      • WIKI
      • WIKI_ALL
      • Math: Wiki lists
      • Glossary of areas of mathematics
      • WIKI_collections
      • Mathematics for Computer Science
      • Mathematics Classification
      • math
      • Resources
      • Math on YouTubel Video Playlists
      • wiki resources
    • 980-hierarchy
      • HIERAR
      • Math: Hierarchy
      • Math HIERARCHY
      • classification
        • Mathematics
        • https://ncatlab.org/nlab/all_pages https://ncatlab.org/nlab/all_pages/reference https://ncatlab.org/
        • Math Classification and Topical Pages
        • Areas of mathematics
        • Areas of mathematics
        • Math Classification: CCS
        • Math hierarchy
        • Computational mathematics
        • Taxonomy: Mathematics
        • Areas of mathematics
        • Mathematics Subject Classification
        • Math fields
        • math-topics
        • Mathematics Subject Classification – MSC
        • MSC Classification Codes
        • mss-top-levels-filenames
        • MSC classification: Top Levels
        • Math classification
    • 990-appendix
      • Math glossary at ENCYCLOPΓ†DIA BRITANNICA
      • Bibliography
      • Math: Abbreviations
      • math.GLOSSARY
    • Math : Canon
      • Main branches of mathematics
      • Enumeration: Math paradigms
      • enum-math-symbols
      • List of mathematical theories
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  1. Relations
  2. Relations

Reflexivity

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Binary relation ⟨x,y⟩ ∈ R

  • Reflexive : βˆ€x. x ∈ X -> ⟨x,x⟩ ∈ R

  • Irreflexive : βˆ€x. x ∈ X -> ⟨x,x⟩ βˆ‰ R

  • Quasi-reflexive : βˆ€xβˆ€y ∈ X. ⟨x,y⟩ ∈ R -> β¦— ⟨x,x⟩ ∈ R ∧ ⟨y,y⟩ ∈ R ⦘

  • Left quasi-reflexivity : βˆ€β€‰x,y ∈ X. ⟨x,y⟩ ∈ R -> ⟨x,x⟩ ∈ R

  • Right quasi-reflexivity: βˆ€β€‰x,y ∈ X. ⟨x,y⟩ ∈ R -> ⟨y,y⟩ ∈ R

  • Coreflexive : βˆ€β€‰x,y ∈ X. ⟨x,y⟩ ∈ R -> x  =  y

Binary relation ~

  • Reflexive : βˆ€x. x ∈ X -> x ~ x

  • Irreflexive : βˆ€x. x ∈ X -> x ≁ x

  • Quasi-reflexive : βˆ€xβˆ€y ∈ X. (x ~ y -> x ~ x ∧ y ~ y)

  • Left quasi-reflexivity : βˆ€β€‰x,y ∈ X. x ~ y -> x ~ x

  • Right quasi-reflexivity: βˆ€β€‰x,y ∈ X. x ~ y -> y ~ y

  • Coreflexive : βˆ€β€‰x,y ∈ X. x ~ y -> x  =  y

Anti-reflexivity (irreflexivity)

A binary relation is called irreflexive or anti-reflexive if it doesn't relate any element to itself. For example the GT relation on ℝ.

Not every relation which is not reflexive is irreflexive; it is possible to define relations where some elements are related to themselves, but others are not ("neither all nor none").

For example, a binary relation "the product of x and y is even" on β„•, is reflexive on the subset of even numbers, irreflexive on the subset of odd numbers, and neither reflexive nor irreflexive on the set on β„• as a whole.

A relation is irreflexive iff its complement is reflexive.

Quasi-reflexivity

A relation ~ on a set X is called quasi-reflexive if every element that is related to some element is also related to itself.

βˆ€xy ∈ X. x ~ y -> (x ~ x ∧ y ~ y)

  • An example is the relation "has the same limit as" on the set of sequences of real numbers: not every sequence has a limit, and thus the relation is not reflexive, but if a sequence has the same limit as some sequence, then it has the same limit as itself.

  • It does make sense to distinguish left and right quasi-reflexivity, defined by βˆ€β€‰x,y ∈ X. x ~ y -> x ~ x and βˆ€β€‰x,y ∈ X. x ~ y -> y ~ y, respectively.

  • For example, a left Euclidean relation is always left, but not necessarily right, quasi-reflexive. A relation R is quasi-reflexive if, and only if, its symmetric closure RβˆͺRT is left (or right) quasi-reflexive.

Coreflexive

A relation ~ on a set X is called coreflexive if for all x and y in X it holds that if x ~ y then x = y

  • An example of a coreflexive relation is the relation on integers in which each odd number is related to itself and there are no other relations.

  • The equality relation is the only example of a both reflexive and coreflexive relation, and any coreflexive relation is a subset of the identity relation.

  • The union of a coreflexive relation and a transitive relation on the same set is always transitive.

  • A relation R is coreflexive if, and only if, its symmetric closure is anti-symmetric.

  • A reflexive relation on a nonempty set X can neither be irreflexive, nor asymmetric, nor antitransitive.

  • The reflexive closure ≃ of a binary relation ~ on a set X is the smallest reflexive relation on X that is a superset of ~

  • Equivalently, it is the union of ~ and the identity relation on X, formally: (≃) = (~) βˆͺ (=). For example, the reflexive closure of (<) is (≀).

  • The reflexive reduction, or irreflexive kernel, of a binary relation ~ on a set X is the smallest relation ≆ such that ≆ shares the same reflexive closure as ~. It can be seen in a way as the opposite of the reflexive closure. It is equivalent to the complement of the identity relation on X with regard to ~, formally: (≆) = (~) (=). That is, it is equivalent to ~ except for where x~x is true. For example, the reflexive reduction of (≀) is (<).

Reflexivity

In mathematics, a binary relation R over a set X is reflexive if it relates every element of X to itself.

I βŠ† R where I is the identity relation on X.

An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.

Properties

  • reflexive

  • non-reflexive

  • irreflexive (anti-reflexive)

  • non-irreflexive

  • coreflexive

  • non-coreflexive

  • quasi-refexive

Examples

  1. Relation "is equal to", i.e. =, on the set of real numbers is reflexive since every real number is equal to itself. Replacing RRR with = we see that xRxxRxxRx becomes x=xx=xx=x, which holds for any xxx.

  2. Let X={1,2,3,4}X=\{1,2,3,4\}X={1,2,3,4}. For relation RRR to be reflefive, it must contain all the identity pairs, R={(1,1),(2,2),(3,3),(4,4)}R=\{(1,1), (2,2), (3,3), (4,4)\}R={(1,1),(2,2),(3,3),(4,4)} of the set X2X^2X2. It can additionally contain any extra pairs, but missing a single id pair will make this relation non-reflexive. If the relation contains only id pairs then this is identity relation.

Quasi-reflexivity

Irreflexivity

A binary relation is irreflexive, or anti-reflexive, if no element is related to itself.

Irreflexive relation prohibits membership for identity pairs. In an endorelation, X2X^2X2, on a set X={1,2,3}X=\{1,2,3\}X={1,2,3}, the relation is irreflexive if it doesn't contain any of the identity pairs i.e. (1,1)(1,1)(1,1), (2,2)(2,2)(2,2) or (3,3)(3,3)(3,3). If it contains even a single id pair, it is non-irreflexive. If it contains all the id pairs it is reflexive.

An example is the "greater than" relation on the set of real numbers, since no real number is greater than itself.

Not every relation which is not reflexive is irreflexive; it is possible to define relations where some elements are related to themselves but others are not.

For example, the binary relation "the product of x and y is even" is reflexive on the set of even natural numbers, irreflexive on the set of odd natural numbers, and neither reflexive nor irreflexive on the set of natural numbers.

Examples of irreflexive relations include:

  • =ΜΈ\not =ξ€ =, "is not equal to"

  • <\lt<, "is less than"

  • >\gt>, "is greater than"

  • βŠ‚\subsetβŠ‚, "is a proper subset of"

https://en.wikipedia.org/wiki/Quasireflexive_relation
https://en.wikipedia.org/wiki/Reflexive_relation