Given two sets a function is any subset of such that each pairs with exactly one. It is denoted as .
It can also be described that a function is a mapping from any element in one set (the domain) to exactly one element in another (the range).
Codomain
The codomain is the set of all possible outputs as declared, not necessarily those actually produced.
In notation , is the codomain.
This is an important distinction especially in functional equations where it might specify that . Every integer must be a valid input and produce an but not every integer has to be obtainable for example is a valid solution despite ever integer except being obtainable.
Range
The range is simply the set of all possible outputs denoted as .
The range is always a subset of the codomain. .
In notation , is the range.
Image
The range is also called the image of the function . You can also say that the image of is .
Preimage
Let the preimage of an element is . Similarly, the preimage of a set is . (Note that the preimage is always a set).
The preimage is defined for all values in the codomain for all functions but a function can have no preimage if it’s not surjective (the preimage is just ) or multiple preimages is it’s not injective.
Functional Properties
Injective
A function is injective on domain if different inputs map to different outputs. Formally denoted as
It can also be state that no two elements in the domain share the same image.
Surjective
is surjective if it covers the whole domain. Formally denoted as
A function is always surjective if .
Bijective
is bijective if it is both *injective and surjective
Indicator Function
Let , the indicator function of denoted by is a function with domain and range defined as