Introduction

Although the concept of functions existed before, it was first formalized in terms of set theory, and described as a particular type of relation between two sets.

Any subset of the Cartesian product between two non-empty sets defines a relation between them.

A particular relation that associates each element of the domain set (DOM) to a single element of codomain set (COD) is called a function.

Cartesian product, relation and function are all sets: fn is a proper subsets of all relations which are subset of Cartesian product:

$\displaystyle \forall a \in A, \forall b \in B \mid aRb = \{(a,b)\}$

$R \ \subseteq \ X \times Y$

$\displaystyle \forall a \in A, \exists !b \in B \mid f(a) = b$

$f \subset R$

$f \subset R \subseteq X \times Y$

All functions are relations, but not all relations are functions: functions ⊂ relations ⊆ dot-product.

relations ⊆ dot-product because the full-relation = dot-product all other relations ⊂ dot-product.

Functions are relations with a set of special properties.