structural:scholar_edition, stylistic:tone, structural:expansion
In mathematics, an injective function is a function f that maps distinct elements of its domain to distinct elements; that is, x1 ≠ x2 implies f(x1) ≠ f(x2). 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 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 injective homomorphism is also called a monomorphism. However, in the more general context of category theory, 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 Homomorphism § Monomorphism for more details. A function f that is not injective is sometimes called many-to-one.