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In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct elements; that is, x1 ≠ x2 implies f(x1) f(x2). (Equivalently, f(x1) = f(x2) implies x1 = x2 in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an is also called a . However, in the more general context of , the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details. A function that is not injective is sometimes called many-to-one. Let be a function whose domain is a set The function is said to be injective provided that for all and in if then ; that is, implies Equivalently, if then in the contrapositive statement. Symbolically, which is logically equivalent to the contrapositive, For visual examples, readers are directed to the gallery section. For any set and any subset the inclusion map (which sends any element to itself) is injective. In particular, the identity function is always injective (and in fact bijective). If the domain of a function is the empty set, then the function is the empty function, which is injective. If the domain of a function has one element (that is, it is a singleton set), then the function is always injective. The function defined by is injective. The function defined by is injective, because (for example) However, if is redefined so that its domain is the non-negative real numbers [0,+∞), then is injective.

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