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

Axioms for the Real Numbers and Integers

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Axioms for the Real numbers

⊛ ⊙ ⊚ ⊗ ⊕ ⊘ ⊜ ⊖ ⊡ ⊞ ⊞ ⊠ ⊡ ⋅ ⊹ ⋅ ⋆ ⋄ ⋇

  1. Existence

    • there exists a set ℝ containing all real numbers, ∃ℝ = {reals}

    • ℝ contains the entire set of integers, ℤ ⊆ ℝ

    • inclusion relation of number sets: ℕ ⊆ ℤ ⊆ ℚ ⊆ ℝ ⊆ ℂ ⊆ ℍ

  2. Closure

    • the set ℝ is closed under addition and multiplication:

    • ∀a,∀b. a,b ∈ ℝ -> a ⊚ b ∈ ℝ, where ⊚ is + or *

  3. Commutativity

    • ∀a,∀b. a,b ∈ ℝ -> a ⊚ b = b ⊚ a, where ⊚ is + or *

  4. Associativity

    • ∀a,∀b,∀c. (a,b,c ∈ ℝ) -> (a ⋇ b) ⋇ c = a ⋇ (b ⋇ c) = a ⋇ b ⋇ c, where ⋇ is + or *

  5. Distributivity

    • left : a ⋄ (b ⊕ c) = (a ⋄ b) ⊕ (a ⋄ c)

    • right: (a ⊕ b) ⋄ c = (a ⋄ c) ⊕ (b ⋄ c)

  6. Zero: 0 is an integer that satisfies a + 0 = a = 0 + a for every real number a.

  7. One is an element that satisfies: ∀a ∈ ℝ . a · 1 = a = 1 · a

  8. Additive inverses: If a is any real number, there is a unique real number −a such that a + (−a) = 0. If a is an integer, then so is −a.

  9. Multiplicative inverses: If a is any nonzero real number, there is a unique real number a−1 such that a · a−1 = 1.

  10. Trichotomy: If a and b are real numbers, then one and only one of the following 3 statements is true: a < b, a = b, or a > b.

  11. Closure of R+: If a and b are positive real numbers, then so are a + b and ab

  12. Addition law for inequalities: If a, b, and c are real numbers and a < b, then a + c < b + c

  13. The well ordering axiom: Every nonempty set of positive integers contains a smallest integer.

  14. The least upper bound axiom: Every nonempty set of real numbers that has an upper bound has a least upper bound.

Properties of Operations

These theorems can be proved from the axioms in the order listed below.

  1. Properties of zero

    • a) a − a = 0

    • b) 0 − a = −a

    • c) 0 · a = 0

    • d) If ab = 0, then a = 0 or b = 0

  2. Properties of signs

    • a) −0 = 0

    • b) −(−a) = a

    • c) (−a)b = −(ab) = a(−b)

    • d) (−a)(−b) = ab

    • e) −a = (−1)a

  3. More distributive properties

    • a) −(a + b) = (−a) + (−b) = −a − b

    • b) −(a − b) = b − a

    • c) −(−a − b) = a + b

    • d) a + a = 2a

    • e) a(b − c) = ab − ac = (b − c)a

    • f) (a + b)(c + d) = ac + ad + bc + bd

    • g) (a + b)(c − d) = ac − ad + bc − bd = (c − d)(a + b)

    • h) (a − b)(c − d) = ac − ad − bc + bd

  4. Properties of inverses (a) If a is nonzero, then so is a−1 (b) 1−1 = 1 (c) (a−1)−1 = a if a is nonzero (d) (−a)−1 = −(a−1) if a is nonzero (e) (ab)−1 = a−1b−1 if a and b are nonzero (f) (a/b)−1 = b/a if a and b are nonzero

  5. Properties of quotients (a) a/1 = a. (b) 1/a = a−1 if a is nonzero. (c) a/a = 1 if a is nonzero. (d) (a/b)(c/d) = (ac)/(bd) if b and d are nonzero. (e) (a/b)/(c/d) = (ad)/(bc) if b, c, and d are nonzero. (f) (ac)/(bc) = a/b if b and c are nonzero. (g) a(b/c) = (ab)/c if c is nonzero. (h) (ab)/b = a if b is nonzero. (i) (−a)/b = −(a/b) = a/(−b) if b is nonzero. (j) (−a)/(−b) = a/b if b is nonzero. (k) a/b + c/d = (ad + bc)/(bd) if b and d are nonzero. (l) a/b − c/d = (ad − bc)/(bd) if b and d are nonzero

  6. Transitivity of inequalities (a) If a < b and b < c, then a < c (b) If a ≤ b and b < c, then a < c (c) If a < b and b ≤ c, then a < c (d) If a ≤ b and b ≤ c, then a ≤ c

  7. Other Properties of inequalities (a) If a ≤ b and b ≤ a, then a = b (b) If a < b, then −a > −b (c) 0 < 1 (d) If a > 0, then a−1 > 0 (e) If a < 0, then a−1 < 0 (f) If a < b and a and b are both positive, then a−1 > b−1. (g) If a < b and c < d, then a + c < b + d. (h) If a ≤ b and c < d, then a + c < b + d. (i) If a ≤ b and c ≤ d, then a + c ≤ b + d. (j) If a < b and c > 0, then ac < bc. (k) If a < b and c < 0, then ac > bc. (l) If a ≤ b and c > 0, then ac ≤ bc. (m) If a ≤ b and c < 0, then ac ≥ bc. (n) If a < b and c < d, and a, b, c, d are nonnegative, then ac < bd. (o) If a ≤ b and c ≤ d, and a, b, c, d are nonnegative, then ac ≤ bd. (p) ab > 0 iff a and b are both positive or both negative. (q) ab < 0 iff one is positive and the other is negative. (r) There is no smallest positive real number. (s) (Density) If a and b are two distinct real numbers, then there are infinitely many rational numbers and infinitely many irrational numbers between a and b

  8. Properties of squares (a) For every a, a2 ≥ 0 (b) a2 = 0 iff a = 0 (c) a2 > 0 iff a > 0 (d) (−a)2 = a2 (e) (a−1)2 = 1/a2 (f) If a2 = b2, then a = ±b (g) If a < b and a and b are both nonnegative, then a2 < b2 (h) If a < b and a and b are both negative, then a2 > b2

  9. Properties of Square Roots (a) If a is any nonnegative real number, there is a unique nonnegative real number √a such that √a2 = a (b) If a = b and a and b are both nonnegative, then √a = √b (c) If a < b and a and b are both nonnegative, then √a < √b (d) If a2 = b and b is nonnegative, then a = ±√b

  10. Properties of Absolute Values

    • a) If a is any real number, then |a| ≥ 0.

    • b) |a| = 0 iff a = 0.

    • c) |a| > 0 iff a 6= 0.

    • d) | − a| = |a|.

    • e) |a| = √a2.

    • f) |a| = max{a, −a}.

    • g) |a−1| = 1/|a| if a 6= 0.

    • h) |ab| = |a| |b|.

    • i) |a/b| = |a|/|b| if b 6= 0.

    • j) |a| = |b| iff a = ±b.

    • k) If a and b are both nonnegative, then |a| ≥ |b| iff a ≥ b.

    • l) If a and b are both negative, then |a| ≥ |b| iff a ≤ b.

    • m) (The triangle inequality) |a + b| ≤ |a| + |b|.

    • n) (The reverse triangle inequality) |a| − |b| ≤ |a − b|

  11. Order properties of integers

    • a) 1 is the smallest positive integer.

    • b) If m and n are integers such that m > n, then m ≥ n + 1

    • c) There is no largest or smallest integer

  12. Properties of Even and Odd Integers

    • In each of the following statements, m and n are assumed to be integers

    • a) n is even iff n = 2k for some int k,

      and odd iff n = 2k + 1 for some int k

    • b) m + n is even iff m and n are both odd or both even

    • c) m + n is odd iff one of the summands is even and the other is odd

    • d) mn is even iff m or n is even

    • e) mn is odd iff m and n are both odd

    • f) n2 is even iff n is even, and odd iff n is odd

  13. Properties of Exponents

    • In these statements, m and n are positive integers

    • a) anbn = (ab)n

    • b) am+n = ambn

    • c) (am)n = amn

    • d) an/bn = (a/b)n if b is nonzero

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