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Injection


Injection

Let f be a function defined on a set A and taking values in a set B. Then f is said to be an injection (or injective map, or embedding) if, whenever f(x)=f(y), it must be the case that x=y. Equivalently, x!=y implies f(x)!=f(y). In other words, f 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 f(x) is injective iff Ker(f)=0.

Bijection

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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Injection

Cite this as:

Weisstein, Eric W. "Injection." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Injection.html

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