Overview
The concept of a function is a central concept in mathematics. Not only is it infused in every branch of mathematics, but it is arguably to central core concept of several different branches.
- Functional Analysis
- Category Theory
- Lambda Calculus and functional programming.
Definition
A function from A to B is a relation (R) from A to B, such that, for each x in A, there is only one y in B where (x,y) in R.
{% \forall x \in A, \exists ! y \; s.t. \, (x,y) \in R %}
Composition
A core distinctive feature of functions is the notion of composition. Given two functions
{% f:A \rightarrow B %}
and
{% g:B \rightarrow C %}
there exists a third function, {% h %}
{% h:A \rightarrow C %}
which is the result of applying first {% f %} and then {% g %}
(that is {% h(a) = g(f(a)) %} )
Types
- One to One - a function is one-to-one if for each element of the range of the function, there is at most one element of the domain that maps to it.
- Onto - a function is onto if every element of the range of a function has at least one element of the domain which maps to it.
- Binary Operation - a binary operation on a set {% S %} is a function from order pairs on the set, {% {s_1,s_2} %} where {% s_i \in S %} into the set {% S %} itself. That is, it takes two arguments are returns a single result which is of the same type as the arguments. Most common algebraic functions such as addition and multiplication are operations.