Let 
be a function
defined on a set 
and taking values in a set 
. Then 
is said to be an injection (or injective map, or embedding) if, whenever 
, it must be the case that 
. Equivalently, 
implies 
. In other words, 
is an injection if it maps distinct
objects to distinct objects. An injection is sometimes also called one-to-one.
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.
See also
Baer's Criterion, Bijection, Domain, Many-to-One, Monomorphism, Range, Surjection
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References
Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). VNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, p. 370, 1989.
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Cite this as:
Weisstein, Eric W. "Injection." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Injection.html