# Injective function

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

More generally, injective partial functions are called partial bijections.