An injection, also called an injective map, embedding, or one-to-one map, is a function 
defined on a set 
and taking values in a set 
such that, whenever 
, it must be the case that 
. Equivalently, 
implies 
. In other words, 
is an injection if it maps distinct
objects to distinct objects.
A linear transformation is injective if the kernel of the function is zero, i.e., a function 
is injective iff 
.
A function which is both an injection and a surjection is said to be a bijection.
In the categories of sets, groups, modules, etc., a monomorphism is the same as an injection, and is used synonymously with "injection" outside of category theory.
