Injective function

In mathematics, an injective function (also known as injection, or one-to-one function^[1] ) is a function f that maps distinct elements of its domain to distinct elements of its codomain; that is, x₁ ≠ x₂ implies f(x₁) ≠ f(x₂) (equivalently by contraposition, f(x₁) = f(x₂) implies x₁ = x₂). In other words, every element of the function's codomain is the image of at most one element of its domain.^[2] 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.^[3] This is thus a theorem that they are equivalent for algebraic structures; see Homomorphism § Monomorphism for more details.

A function ${\displaystyle f}$ that is not injective is sometimes called many-to-one.^[2]