One-to-one function
In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence (a.k.a. bijective function), which uniquely maps all elements in both domain and codomain to each other (see figures).
Occasionally, an injective function from X to Y is denoted f : X ↣ Y, using an arrow with a barbed tail (U+21A6 ↦ RIGHTWARDS ARROW WITH TAIL). The set of injective functions from X to Y may be denoted YX using a notation derived from that used for falling factorial powers, since if X and Y are finite sets with respectively m and n elements, the number of injections from X to Y is nm (see the twelvefold way).
A function f that is not injective is sometimes called many-to-one. However, the injective terminology is also sometimes used to mean "single-valued", i.e., each argument is mapped to at most one value.
A monomorphism is a generalization of an injective function in category theory.
Let f be a function whose domain is a set X. The function f is said to be injective provided that for all a and b in X, whenever f(a) = f(b), then a = b; that is, f(a) = f(b) implies a = b. Equivalently, if a ≠ b, then f(a) ≠ f(b).
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