A function is injective (one-to-one) if every possible element of the codomain is mapped to by at most one argument. Equivalently, a function is injective if it maps distinct arguments to distinct images. An injective function is an injection. The formal definition is the following.
- The function is injective iff for all , we have
- A function f : A → B is injective if and only if A is empty or f is left-invertible; that is, there is a function g : f(A) → A such that g o f = identity function on A. Here f(A) is the image of f.
- Since every function is surjective when its codomain is restricted to its image, every injection induces a bijection onto its image. More precisely, every injection f : A → B can be factored as a bijection followed by an inclusion as follows. Let fR : A → f(A) be f with codomain restricted to its image, and let i : f(A) → B be the inclusion map from f(A) into B. Then f = i o fR. A dual factorisation is given for surjections below.
- The composition of two injections is again an injection, but if g o f is injective, then it can only be concluded that f is injective. See the figure at right.
- Every embedding is injective.
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