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In mathematics, an injective function is a function which associates distinct arguments to distinct values. More precisely, a function f is said to be injective if, for every y in the codomain, there is at most one x in the domain such that f(x) = y. Put another way, f is injective if f(a) = f(b) implies a = b (or a b implies f(a) f(b)), for any a, b in the domain. An injective function is called an injection, and is also said to be an information-preserving or one-to-one function. (However, this name is best avoided, since some authors understand it to mean a one-to-one correspondence, i.e. a bijective function.) A function f that is not injective is sometimes called many-to-one. However, this name too is best avoided, since it is sometimes used to mean "single-valued" — i.e. each argument is mapped to at most one value.
Examples and counter-examples More generally, when X and Y are both the real line R, then an injective function f : R → R is one whose graph is never intersected by any horizontal line more than once. Injections are invertible Another definition of injection is a function whose effect can be undone. More precisely, f : X → Y is injective if there exists a function g : Y → X such that g(f(x)) = x for every x in ´ X; that is, g o f equals the identity function on X. Note that g may not be a complete inverse of f because the composition in the other order, f o g, may not be the identity on Y. In fact, to turn an injective function f : X → Y into a bijective (hence invertible) function, it suffices to replace its codomain Y by its actual range J = f(X). That is, let g : X → J such that g(x) = f(x) for all x in X; then g is bijective. Indeed, f can be factored as inclJ,Yog, where inclJ,Yis the inclusion function from J into Y. Other properties Category theory view In the language of category theory, injective functions are precisely the monomorphisms in the category of sets. See also | ||||||||
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