Onto or Surjective Functions
Suppose we have two sets
and
We can define a function acting on the elements ofwhich sends those elements ofonto unique – needed so that f is a function, since a function cannot be one to many - elements ofIf it is the case that for every element
of
there exists some
in such that
then
is said to be a function fromontoIn this caseis said to be a surjective function or surjection.
Note thatdoes not need to be one to one to be surjective. We could take
and
for all
thenis onto
The above functions are discrete, but we can define function that act on intervals.
is not onto
because there is no
for which
If instead we defineas
thenis onto. We can always make a function onto some subset ofin this way because by definition, a function is onto it's image.