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 of its codomain; that is, x1 ≠ x2 implies f(x1) f(x2) (equivalently by contraposition, implies x1 = x2 ). In other words, every element of the function's codomain is the image 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 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 for more details. A function f that is not injective is sometimes called many-to-one.

Definition

Let f be a function whose domain is a set X. The function f is said to be injective provided that for all a and b in X, if then a = b; that is, f(a) = f(b) implies a=b. Equivalently, if a \neq b, then in the contrapositive statement. Symbolically, which is logically equivalent to the contrapositive, An injective function (or, more generally, a monomorphism) is often denoted by using the specialized arrows ↣ or ↪ (for example, or ), although some authors specifically reserve ↪ for an inclusion map.

Examples

For visual examples, readers are directed to the gallery section. More generally, when X and Y are both the real line \R, then an injective function is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the.

Injections can be undone

Functions with left inverses are always injections. That is, given if there is a function g : Y \to X such that for every x \in X, g(f(x)) = x, then f is injective. In this case, g is called a retraction of f. Conversely, f is called a section of g. Conversely, every injection f with a non-empty domain has a left inverse g. It can be defined by choosing an element a in the domain of f and setting g(y) to the unique element of the pre-image f^{-1}[y] (if it is non-empty) or to a (otherwise). The left inverse g is not necessarily an inverse of f, because the composition in the other order, f \circ g, may differ from the identity on Y. 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.

Injections may be made invertible

In fact, to turn an injective function f : X \to Y into a bijective (hence invertible) function, it suffices to replace its codomain Y by its actual image J = f(X). That is, let g : X \to J such that g(x) = f(x) for all x \in X; then g is bijective. Indeed, f can be factored as where is the inclusion function from J into Y. More generally, injective partial functions are called partial bijections.

Other properties

Proving that functions are injective

A proof that a function f is injective depends on how the function is presented and what properties the function holds. For functions that are given by some formula there is a basic idea. We use the definition of injectivity, namely that if then x = y. Here is an example: Proof: Let Suppose So implies 2 x = 2 y, which implies x = y. Therefore, it follows from the definition that f is injective. There are multiple other methods of proving that a function is injective. For example, in calculus if f 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. In linear algebra, if f is a linear transformation it is sufficient to show that the kernel of f contains only the zero vector. If f 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 f of a real variable x is the horizontal line test. If every horizontal line intersects the curve of f(x) in at most one point, then f is injective or one-to-one.

Gallery