🗄️
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
      • Gödel'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 › 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-Gödel 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
      • Gödel'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 λ-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: η-conversion
    • lambda-calculus
      • Alonzo Church
      • Inference rules for lambda calculus
      • Lambda Calculus: Introduction
      • Lambda abstraction
      • Lambda application
      • Lambda Calculus: Definition
      • About λ-calculus
      • Type inference
      • Lambda Calculus
      • Lambda Calculus: Introduction
      • Introduction to λ-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 › Probability theory: Glossary
      • Statistics › Probability theory › Topics
      • Statistics › Probability theory › 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 ENCYCLOPÆDIA BRITANNICA
      • Bibliography
      • Math: Abbreviations
      • math.GLOSSARY
    • Math : Canon
      • Main branches of mathematics
      • Enumeration: Math paradigms
      • enum-math-symbols
      • List of mathematical theories
      • enum-algebras
        • Group-like algebraic structures
        • Group
        • Groupoid
        • magma
        • Monoid
        • Semigroup
      • enum-axioms-indices
        • 03-variants
        • axioms-relations
        • disjoint-sets
        • ordered-pair
      • enum-axioms
        • Axiom of Extensionality
        • Axioms
          • List of Axioms
          • List of Axioms
          • LINKS
          • Index of axioms
          • Index of axioms
          • List of Axioms
          • List of Axioms in Relations
          • _laws
          • Index of axioms
          • Index of properties
          • Axioms in Boolean Algebra
          • Axioms for the Real Numbers and Integers
          • Axioms for the Real Numbers and Integers
          • ZF Axioms
          • ZFC Axioms
          • Axioms in logic
          • Axioms in set theory
          • List of axioms in set theory
        • Axioms
          • Absorption
          • Annihilation
          • Associativity
          • Cancellation
          • Closure
          • Commutativity
          • Complement laws
          • Distributivity
          • Domination
          • Idempotency
          • identity
          • Invertibility
          • involution
          • Linearity
          • Monotonicity
          • Well-definedness
          • Well-formedness
          • Well-foundedness
          • Well-ordering
      • enum-inference-rules
        • Enumeration: Rules of Inference
        • Enumeration: Rules of Inference for Intuitionistic Natural Deduction
      • Math › Logic formulae
        • Logic formulas concerning functions
        • Logic formulas concerning orders
        • Logic formulas concerning relations
        • Logic formulas concerning sets
      • enum-logic-system-elements
        • Conjecture
      • enum-logic-system-properties
        • Completeness
        • Consistency
        • Decidability
      • enum-topical-hierarchies
        • Outline of category theory
    • Mathematical Lists and Indices
      • Index of algebraic structures
      • Math equations and formulas
      • The fundamental number sets
      • index-of-math-elements
      • Arity
      • List of binary relations
      • Pretty formulae
      • Index of integer sequences
      • List of conjectures
      • Mathematical objects
      • Theorems
      • List of mathematical theories
      • list-of-unsolved-problems
      • Lists of problems in math
      • People
      • list-of-quotes-and-phrases
    • Mathematical elements
      • Mathematical (elements): PAGES
      • Mathematical abstraction
      • Mathematical analysis
      • Mathematical argumentation
      • Mathematical axiom
      • Mathematical concept
      • Mathematical constant
      • Mathematical deduction
      • Mathematical definition
      • Mathematical element
      • Mathematical expression
      • Mathematical formula
      • Mathematical method
      • Mathematical model
      • Mathematical notation
      • Mathematical object
      • Mathematical primitive
      • Mathematical reasoning
      • Mathematical statement
      • Mathematical structure
      • Mathematical symbol
      • Mathematical synthesis
  • debrief-name: math section-code: 800 section-name: misc section-desc: Misc and unclassified mathemat
    • 810-statistics
      • STATISTICS: HIERARCHY
      • STATISTICS: TERMS
      • Statistics › Topics
      • Statistics WIKI LINKS
      • Mathematical Concepts
      • Linear Regression
    • 888-calculus
      • Calculus
      • Math › Calculus › TERMS
Powered by GitBook
On this page
  • Details
  • In programming languages
  • Associative ops

Was this helpful?

  1. debrief-name: math section-code: 900 section-name: aggregations section-desc: Aggregations, indices,
  2. Math : Canon
  3. enum-axioms
  4. Axioms

Associativity

PreviousAnnihilationNextCancellation

Last updated 4 years ago

Was this helpful?

Associativity determines how an operation behaves in relation to itself when multiple occurances of the same operator occur in the same expression free of parenthesis. On the other hand, commutativity that determines how an operator behaves in relation with other operators.

Operator associativity:

  • left-associative

  • right-associative

  • fully-associative, total associativity

  • non-associative

  • associates to the left, e.g. function application e.g. λa.λb.λc.λd.abcd = λa.λb.λc.λd.((ab)c)d

  • right-associative, associates to the right, e.g. function abstraction

    e.g. λa.λb.λc.λd.abcd = λa.(λb.(λc.(λd.abcd)))

  • associative, fully-associative,

    e.g.

    p ∧  q  ∧  r  ∧ s = 

    ((p ∧ q) ∧ r) ∧ s =

    p ∧ (q  ∧ (r  ∧ s)) = 

    (p ∧ q) ∧ (r ∧ s) =

    p ∧ (q  ∧  r) ∧ s = 
    p ∧  q  ∧  r  ∧ s
  • non-associative

In math, the associative property is a quality of a large number of infix operators. Positioned in between the operands, it is their placement that makes the order of evaluation uncertain with two or more occurances of the same operator in a row.

In general, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a mathematical expression.

These conventions exist to eliminate ambiguity while allowing notation to be as brief as possible. Where it is desired to override the precedence conventions, or even simply to emphasize them, parentheses ( ) can indicate an alternative order or reinforce the default order to avoid confusion.

The order of operations, which is used throughout mathematics, science, technology and is:

  • groupings

  • powers: exponentiation and n-root

  • multiplication and division

  • addition and subtraction

  • otherwise, left to right

In exponentiation, exponents of exponents (tower of exponents) are evaluated in bottom-associative (bottom-up) order, 2^3^2^4 = ((2^3)^2)^4

have a special treatment: they go top-down (do the exponent at the top first).

The commutative and associative laws of addition and multiplication allow adding terms in any order, but mixed operations must obey the standard order of operations.

Functions use parentheses around the argument list to avoid ambiguity, but they may be omitted when the argument is atomic, e.g. sin π.

Grouping symbols are used to overide or emphasize evaluation order. Symbols of grouping can be removed using the associative and distributive laws, also they can be removed if the expression inside the symbol of grouping is sufficiently simplified so no ambiguity results from their removal. A horizontal fractional line doubles as a symbol of grouping, and so does the vinculum (prolongated bar that runs over a radicand in a √ expression).

The root symbol √ is traditionally prolongated by a bar (called vinculum) over the radicand (this avoids the need for parentheses around the radicand).

the order of evaluation of operations in an expression depends on an arbitrarily-assigned, possibly explicit, precedence level. When implicit, the precedence depends on the previusly established set of rules.

Arithmetic operations are arranged in a less-precedent-then and more-precedent-then orders.

+ ⊏ - ⊏ ⨯ ⊏ ÷ ⊏ ^ ⊏ ⇈

P - Parentheses

E - Exponents

M - Multiplication

D - Division

A - Addition

S - Subtraction

have precedence in terms of a greater or lesser precedence in relation to another operator, such that + has less precedence then *

¬p ∧ q

we are mostly concerned with the associativity of a polyadic (arity > 0) infix operator.

mostly brought in connection with binary operators, although operators of other arity may also express it.

Within an expression containing two or more occurrences in a row of the same associative operator,

the order in which the operations are performed does not matter as long as the sequence of the operands is not changed.

That is, rearranging the parentheses in such an expression will not change its value.

associative operations

In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.

Associativity is a property of a binary operation (and the underlying, carrier set) that makes the order, in which the elements of the set are combined, immaterial.

  p ∧  q  ∧  r  ∧ s = 
((p ∧  q) ∧  r) ∧ s =
  p ∧ (q  ∧ (r  ∧ s)) = 
 (p ∧  q) ∧ (r  ∧ s) = 
  p ∧ (q  ∧  r) ∧ s = 
  p ∧  q  ∧  r  ∧ s

∀x,y,z . x⋆(y⋆z)=(x⋆y)⋆z=x⋆y⋆z\forall x,y,z\ .\ x \star (y \star z) = (x \star y) \star z = x \star y \star z∀x,y,z . x⋆(y⋆z)=(x⋆y)⋆z=x⋆y⋆z

For all elements x,y,z in a carrier set, whether we first combine x and y, the result of which we then combine with z, is the same as combining a with the result of combining y and z. Since the order of operations does not matter, we can dispense with the parenthesis.

Details

Associativity is sometimes an axiom, sometimes a law, a principle, a notion, but always a property of a mathematical entity (entity is a math object or concept/notion in the most abstract sense, a generic, catch-all term).

Associativity is a property of an operation. In fact, an operation that relates two terms, so it's gotta be a binary operation, and it arises when combining at least 3 terms.

In algebra, a binary operation, denoted by ⊙, is associative (has associative property) iff the order in which the terms are combined is immaterial.

Who does, in fact, posses associativity? An operation? Terms? Terms and the operation together? With many other axioms it is always the latter: we have a set/structure/type (called carrier set in abstract algebra) whose elements are combined with a binary operation, and these two (set and operation) together form, say, an algebraic structure iff they uphold a set of prescibed axioms.

So, for example, a monoid is an algebraic structure, and many things are monoids which means there is (i) a carrier set (ii) a binary operation (iii) and the axioms of closure, associativity and identity are respected. By whom? By the carrier set (also a type in programming languages) and the associated binary operation together. This is why it is said that, e.g., (i) the set of natural numbers under (ii) addition forms a (iii) monoid (the prescribed axioms come and are enforced by the monoid).

In arithmetics, the plus operator is associative no matter what kind of numbers you through at it, so with some axioms apparently and in some situations, it could be said that the op itself has a certian property, like associativity. However, it's better to think about both, the carrier set and operation together eminating such property.

The associative property is a property of some binary operations. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.

a ⊙ b ⊙ c = (a ⊙ b) ⊙ c = a ⊙ (b ⊙ c)

In programming languages

the associativity of an operator is a property that determines how operators of the same precedence are grouped in the absence of parentheses. If an operand is both preceded and followed by operators (for example, ^ 3 ^), and those operators have equal precedence, then the operand may be used as input to two different operations (i.e. the two operations indicated by the two operators).

The choice of which operations to apply the operand to, is determined by the associativity of the operators.

Operators may be:

  • associative operations can be grouped arbitrarily

  • left-associative operations are grouped from the left

  • right-associative operations are grouped from the right

  • non-associative operations cannot be chained, often because the output type is incompatible with the input types.

The associativity and precedence of an operator is a part of the definition of the programming language; different programming languages may have different associativity and precedence for the same type of operator.

The concept of notational associativity is related to, but different from the mathematical associativity.

An operation that is mathematically associative, by definition, requires no notational associativity. For example, addition has the associative property, therefore it does not have to be either left-associative or right-associative.

An operation that is not mathematically associative, however, must be notationally left-, right-, or non-associative. For example, subtraction does not have the associative property, therefore it must have notational associativity.

Associative ops

x + y + z   addition
x * y * z   multiplication

h ∘ g ∘ f   function composition

p ∧ q ∧ r   logical conjunction
p ∨ q ∨ r   logical disjunction
https://en.wikipedia.org/wiki/Associative_property
https://en.wikipedia.org/wiki/Operator_associativity