Applications
Let
be sets. A function
assigns to each
a unique element
. Functions are also called maps. mappings or application.
Definition :
Let
be a function. Then
is called the domain of
and
is called the codomain of
. We write
to indicate that is the function that maps
to
.
Example :
Let
and
. Then we can define a function
by setting
and
.
is the image of
inder
,
is the perimage of
under
.
Injective function
The function is injective, if each element of the codomain is mapped to by at most one element of the domain, or equivalently, if distinct elements of the domain map to distinct elements in the codomain.
Definition :
A function
is injective if we have:
. An injection is also know as a one to one function.
Example :
The function
defined by
is not injective since
The function
defined by
is injective.
Surjective function
The function is surjective, if each element of the codomain is mapped to by at least one element of the domain. That is, the image and the codomain of the function are equal.
Definition :
A function
is surjective if we have
.
Example :
The function
is surjective.
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