Function Theory: TOPIC

• Function Theory

• Function definition

  • function definition

  • function as a set

  • function as a relation

  • function as a special relation

  • left-unique relation

  • right-serial relation

  • function as a rule

  • function as a process

  • function is a set

  • function is a set of ordered pairs

  • as a graph

  • as a rule (procedure)

• Function components

  • domain

  • codomain

  • range

  • image

  • pre-image

• Function properties

  • injectivity

  • surjectivity

  • bijectivity

  • totality

  • total function

  • partial function

  • primitive function

  • one-to-one function

  • onto function

  • surjection

  • injection

  • bijection

  • one-to-one function

  • one-to-one correspondence

  • map

  • mapping

  • transformation

  • morphism

• Function types

  • total

  • partial

  • identity function

  • invertible function

  • constant function

• Function concepts

  • fixpoint

• Function in programming

  • named function

  • anonymous function, x ↦ f (x)

• Functions in computability theory

  • Turing-computable function

  • μ-recursive functions

  • Computable function

  • PRF

  • GRF

• Functions: alt

  • map, mapping

  • transformation

  • morphism

  • arrow

• Morphisms

  • homomorphism

  • homeomorphism

  • endomorphism

  • hylomorphism

  • catamorphism

• Examples by domain and codomain

  • X → 𝔹, 𝔹 → X, 𝔹ⁿ → 𝔹

  • X → ℕ, ℕ → X

  • X → ℤ, ℤ → X

  • X → ℝ, ℝ → X, ℝⁿ → X

  • X → ℂ, ℂ → X, ℂⁿ → X

• Constructions  

  • restriction

  • composition

  • inverse

• Generalizations  

  • partial function

  • multivalued function

  • implicit function

• Lambda Calculus formalisam of functions function abstraction function application λ-calculus lambda calculus function abstraction function application alpha equivalence beta conversion eta conversion

• function notation function notation arrow notation index notation dot notation specialized notations

• function specification by listing function values by a formula inverse function implicit function using differential calculus by recurrence relation

• representing a function set of ordered pairs as a formula graphs and plots table bar chart

  • standard functions

  • composition

  • image and preimage

  • injective function

  • surjective function

  • bijective function

  • restriction and extension

  • multivariate function

  • calculus

    • real function

    • vector-valued function

  • misc

    • function space

    • multi-valued functions

    • in the foundations of mathematics and set theory

    • generalizations