functions_math.theory.txt

                                  ┏━━━━━━━━━━━━━━━━━━━━┓
                                  ┃   FUNCTIONS_MATH   ┃
                                  ┗━━━━━━━━━━━━━━━━━━━━┛

n-ary relation

Binary relation:
  - notation:
     - R S ...
  - rule assigning some x (where x∈ X) to some y (where y∈ Y)
  - terms:
     - domain: X
     - codomain: Y
     - image of x: y
     - preimage of y: x
     - image of R: set of all y with a preimage
        - noted Im(R)

Binary relations can be defined:
  - with algebraic rule:
     - e.g. f: X -> X^2
     - can include logical conditions
  - with list of images|preimages:
     - can be:
        - numerical: as a table
        - graphical: as a graph
     - can interpolate

Domain:
  - set can be defined either:
     - with explicit elements
     - with a logical condition for any elements
        - named natural domain
  - support: subset of all x with an image != 0

Uniqueness:
  - right-uniquess + left-uniquess
  - right-uniqueness:
     - also named functionality
     - when no preimages has multiple images
  - left-uniqueness:
     - also named injectivity or one-to-one (1-1)
     - when no image has multiple preimages

Totality:
  - left-totality + right-totality
  - left-totality:
     - also named seriality
     - when no preimages has 0 images
  - right-totality:
     - also named surjectivity or onto
     - when no image has 0 preimages
     - i.e. Im(f) = Y
     - g: domain(f) -> Im(f) is always right-total

Bijectivity
  - uniquess + totality
  - every image has exactly 1 preimage
  - implies |X| = |Y|
     - if |X| = |Y|, left-total + right-unique => right-total + left-unique (and vice-versa)

Partial function:
  - binary function that is right-unique
  - notation:
     - f: X ->/ Y
     - f: X ⇸ Y

Function:
  - partial function that is left-total
  - notation:
     - f: X -> Y (whole domain)
     - f(x) = y (specific preimage)
     - F G ... (for function composition)

Function composition:
  - notation: F ∘ G
  - means g(f(x))

Functional power:
  - notation:
     - Fⁿ or f^n
     - fⁿ(x) or f^n(x)
        - careful: is also an alternative notation for derivatives f''(x)
  - means F ∘ F ∘ ...

Function space:
  - noted [X -> Y]
  - set of all possible f: X -> Y

Operations:
  - also named operators
  - function where preimages|images are tuples
  - notation:
     - f: X * Y * ... -> Z
        - i.e. domain is cartesian product of X * Y * ...
     - X OP Y
  - notation for unspecified|unknown operation
     - X • Y
     - XY (juxtaposition)
  - operands:
     - other name for preimages
     - also named inputs or arguments
  - outputs
     - other name for images
     - also named return values
  - arity:
     - also named rank or adicity
     - number of operands
     - names:
        - nullary|niladic: 0
        - unary|monadic: 1
        - binary|dyadic: 2
        - ternary|triadic: 3
        - quaternary|tetradic: 4
        - multary|multiary|polyadic: 2+
        - N-ary: N
        - variadic|multigrade|anadic: indefinite
  - often has own notation:
     - often uses a sign:
        - can be written:
           - infix: between operands
           - prefix|postfix: before|after operands
     - precedence:
        - also named order of operations
        - decides how to group operands and operations in infix notation
           - by assigning a higher|lower precedence to each operation

Identity element:
  - also named only identity
  - notation:
     - e
     - 1ₛ (for set S)
     - id
  - y so that:
     - left-identity: f(y, x) <=> x
     - right-identity: f(x, y) <=> x
     - two-sided identity: both
  - there can be several left|right identities
  - there cannot be both left|right identities, except for one single two-sided identity
     - otherwise f(x,y) <=> x || y, if both are different identity elements
  - name for specific operations:
     - additive identity: addition
     - multiplicative identity: multiplication
        - also named unity

Unity
  - adjective: unital
  - also named identity property
  - when for any x, there is an identity element y

Identity function:
  - also named identity relation|map|transformation
  - noted idₓ (for domain x)
  - g: x -> x
  - identity element of function composition
  - involutory

Absorbing element
  - also named zero|annihilating element
  - element x, for a binary function f:
     - left-absorbing|zero: ∀ y. f(x,y) = x
     - right-absorbing|zero: ∀ y. f(y,x) = x
     - absorbing|zero: both

Inverse:
  - inverse element:
     - also named only inverse
     - with identity element id, y so that:
        - right-inverse: f(x, y) = id
        - left-inverse: f(y, x) = id
        - two-sided inverse: both
     - y is [left|right-]inverse element
     - x is [left|right-]invertible element
        - also named unit
     - if there are several left|right-identities, there can be several left|right-inverses
  - name for specific operations:
     - additive inverse: addition
        - also named opposite|negation|sign change
     - multiplicative inverse: multiplication
        - also named reciprocal
  - invertibility of a set:
     - when all elements are invertible
  - implied by:
     - all of:
        - cancellation
        - unity
        - right-totality for a given x
     - reason: bijection from Y to Z, i.e. exactly one z must be identity element
  - formulas:
     - f(x, f(x, y)) = x

Inverse function:
  - also named anti-function
  - noted f⁻¹ or f^-1
  - inverse element of function composition
     - i.e. f⁻¹ is inverse of f if:
        - right-inverse: f ∘ f⁻¹ = idₓ
           - i.e. f⁻¹(f(x)) = x
        - left-inverse: f⁻¹ ∘ f = idₓ
           - i.e. f(f⁻¹(x)) = x
        - inverse: both
  - partial inverse function:
     - also named quasi-inverse
     - when f is not right-total
     - i.e. only Im(f) can be inverted, not whole codomain
  - [quasi-]invertibility of a function:
     - when it has an [quasi-]inverse function
     - equivalent to bijectivity
        - for quasi-inverse: only to left-uniqueness
  - inherits from function:
     - left|right-unique -> right|left-unique
     - left|right-total -> right|left-total
     - involution
     - idempotence
     - closure
  - formulas:
     - (f⁻¹)⁻¹ = f
     - (f ∘ g)⁻¹ = g⁻¹ ∘ f⁻¹

Algebraic division:
  - inverse function of a binary function
     - binary function often noted *
  - notation for left division:
     - / (solidus)
     - ÷ (obelus, division sign)
     - :
     - horizontal fraction bar
  - notation for right division:
     - \
  - in x/y:
     - x is dividend|numerator
     - y is divisor|denominator
  - if multiplication x*y = z is not commutative, there is:
     - right division: z/y = x
     - left division: x\z = y
  - inherits from function:
     - unity
  - never:
     - absorbing element
     - commutative
     - associative:
        - is left-associative instead:
           - x/y/z = (x/y)/z
           - x\y\z = (x\y)\z
     - alternative
     - power-associative
     - flexible
     - medial
     - distributivity

Cancellation property:
  - also named divisibility
  - possibility to do algebraic division
  - of an element x, for x*y = z:
     - can be:
        - left-cancellative: z/x = y
        - right-cancellative: z/y = x
        - cancellative: both
     - equivalent to either:
        - left-uniquess
        - x*y = x*z => y = z
           - or y*x ...
  - of a set: if all elements are cancellative
  - formulas:
     - x = y/(y/x)
     - if associativity of *
        - x/(y*z)     = (x/y)/z
        - x/(y/z)     = (x*z)/y
        - x*(y/z)     = (x*y)/z
        - (x/y)*(v/w) = (x*v)/(y*w)
        - x/y         = (x*w)/(y*w)
        - if commutativity of *
           - x*(y/z)  = y*(x/z)
        - if unity of *
           - x/y      = id/(y/x)
           - x/y      = x*(id/y)
           - x/(y/z)  = x*(id/(y/z))

Involution:
  - adjective: involutory
  - f(f(x)) = x
     - i.e. f = f⁻¹

Idempotence:
  - f(f(x)) <=> f(x)

Nilpotence:
  - x where ∃ n. x^n = 0, n∈ N
     - n is named index|degree
  - unipotence: x + 1

Commutativity:
  - for binary relations, named "symmetry"
     - opposite: "antisymmetry"
  - f(x,y) <=> f(y,x)

Associativity:
  - also named association
  - g over f:
     - f(x, g(y,z)) <=> g(f(x,y), z)
  - f over g:
     - f(g(x,y), z) <=> g(x, f(y,z))
  - left|right-associativity:
     - when not associative and operations must be evaluated from left|right-to-right|left

Alternativity:
  - associativity when 2 operands (including middle) are equal
  - can be:
      - left-alternative: f(x, g(x,y)) <=> g(f(x,x), y)
      - right-alternative: f(x, g(y,y)) <=> g(f(x,y), y)
      - alternative: both
  - implied by associativity

Flexibility:
  - associativity when 2 operands (excluding middle) are equal
  - f(x, g(y,x)) <=> g(f(x,y), x)
  - implied by either associativity or commutativity

Power-associativity:
  - associativity when 3 operands are equal
  - f(x, g(x,x)) <=> g(f(x,x), x)
  - implied by either alternativity, flexibility or idempotency
  - if function is multiplication:
     - means unique result of x^n, where n∈ N

Distributivity:
  - is:
     - left-distributivity: f(x, g(y,z)) <=> f(g(x,y), g(x,z))
     - right-distributivity: f(g(y,z), x) <=> f(g(y,x), g(z,x))
     - [auto-]distributivity: both
  - if f commutative, left-distributivity same as right-distributivity
  - formulas (using generic * + / for f g f⁻¹)
     - (x+y)*z         = x*z + y*z
     - (x+y)*(v+w)     = x*v + x*w + y*v + y*w (factorization)
     - if cancellation of *
        - (x+y)/z      = x/z + y/z
        - if associativity of *
           - x/y + v/w = (x*w + v*y)/(y*w)

Absorption:
  - left: f(x, g(x,y)) = x and
          g(x, f(x,y)) = x
  - right: f(g(x,y), y) = y and
           g(f(x,y), y) = y

Medial:
  - also named abelian, alternation, transposition, interchange, bi-commutativity, bisymmetry, surcommutativity, entropy
  - f(f(w,x), f(y,z)) <=> f(f(w,y), f(x,z))
  - implied by commutativity + associativity
  - implies semimedial:
     - left-semimedial: f(f(x,x), f(y,z)) <=> f(f(x,y), f(x,z))
     - right-semimedial: f(f(w,x), f(y,y)) <=> f(f(w,y), f(x,y))
     - semimedial: f(f(x,x), f(y,y)) <=> f(f(x,y), f(x,y))

Reflexivity:
  - f(x,x) = true
  - opposite: "irreflexivity"

Transitivity:
  - f(x,y) && f(y,z) => f(x,z)

Equivalence:
  - symmetry + reflexivity + transitivity

Extensivity:
  - x ⊆ f(x)

Monotony:
  - increasing:
     - also named non-decreasing
     - weakly: x <= y => f(x) <= f(y)
     - strictly: x < y => f(x) < f(y)
  - decreasing:
     - also named non-increasing
     - weakly: x >= y => f(x) >= f(y)
     - strictly: x >= y => f(x) >= f(y)

Closure property:
  - when domain is either same set of power set of codomain
  - i.e. f: S [* S [*...]] -> S
  - closed set:
     - smallest S closed under f
  - closure operator:
     - also named hull|consequence operator
     - f: S -> T
        - where S ⊆ T
        - for a given function g
     - returns closed set T under g
     - properties:
        - idempotent
        - monotone
        - extensive
  - closure space: closed set + closure operator

Representations:
  - functional diagram:
     - domain|codomain: left|right
     - images|preimages: dots
     - image|preimage relations: arrows between them
  - graph:
     - domain|codomain: abscissa|ordinate (x|y axis)
     - images|preimages: lines on abscissa|ordinate
     - image|preimage relations: dots at the intersection of images|preimages lines
        - x|y-intercept: dot where x|y is 0

Indicator function:
  - also named characteristic function, or boolean predicate function
  - f: X -> {0,1} with 1 if X∈ Y, 0 if X∉ Y

Mathematical model:
  - function used to represent a sytem (often in real life)
  - can be:
     - analytic: based on rule or logical condition
     - curve-fitting: based on list of images|preimages (regression)