Let 
be a function
defined on a set 
and taking values in a set 
. Then 
is said to be a surjection (or surjective map) if, for any
, there exists an 
for which 
. A surjection is sometimes referred to as being "onto."
Let the function be an operator which maps points in the domain to every point in the range
and let 
be a vector space with 
. Then a transformation 
defined on 
is a surjection if there is an 
such that 
for all 
.
In the categories of sets, groups, modules, etc., an epimorphism is the same as a surjection, and is used synonymously with "surjection" outside of category theory.
See also
Bijection, Domain, Epimorphism, Injection, Many-to-One, Range
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Cite this as:
Weisstein, Eric W. "Surjection." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Surjection.html