uni_notes/DAS/Functions.md

A function takes in an element from its domain, transforms it in someway and outputs that transformed element, which is part of the co-domain. Every element of the domain must map to a value in the co-domain, all values of the co-domain that are mapped to form the functions image. A function must be deterministic - one input can only map to a single output.

Notation

General function notation: f: X -> Y

[!INFO] f: name of the function X: Domain Y: Co-domain f(x): Image of f X, f(x) and Y are Set Theory

For any x in X the output f(x) is an element of Y.

Mapping Properties

Injectivity

A function is injective if every element in Y has at most one matching x in X.

• forall y in Y,exists^(<=1) x in X : f(x) = y

Surjectivity

A function is surjective if every element y in Y has at minimum one matching x in X

• forall y in Y, exists x in X : f(x) = y

Bijectivity

A function is bijective if every element y in Y has exactly one matching x in X (it is injective and surjective)

• forall y in Y, exists excl x in X : f(x) = y