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.

that is not injective is sometimes called many-to-one.

be a function whose domain is a set

which is logically equivalent to the contrapositive,[4]

An injective function (or, more generally, a monomorphism) is often denoted by using the specialized arrows ↣ or ↪ (for example,

), although some authors specifically reserve ↪ for an inclusion map.

[5] For visual examples, readers are directed to the gallery section.

is one whose graph is never intersected by any horizontal line more than once.

This principle is referred to as the horizontal line test.

[2] Functions with left inverses are always injections.

with a non-empty domain has a left inverse

It can be defined by choosing an element

to the unique element of the pre-image

In other words, an injective function can be "reversed" by a left inverse, but is not necessarily invertible, which requires that the function is bijective.

In fact, to turn an injective function

into a bijective (hence invertible) function, it suffices to replace its codomain

For functions that are given by some formula there is a basic idea.

There are multiple other methods of proving that a function is injective.

is a differentiable function defined on some interval, then it is sufficient to show that the derivative is always positive or always negative on that interval.

is a linear transformation it is sufficient to show that the kernel of

is a function with finite domain it is sufficient to look through the list of images of each domain element and check that no image occurs twice on the list.

A graphical approach for a real-valued function

is the horizontal line test.

If every horizontal line intersects the curve of