injective function: Information and Much More from Answers.com

"Injective" redirects here. For injective modules, see Injective module.

In mathematics, an injective function is a function which associates distinct arguments to distinct values. More precisely, a function f is said to be injective if it maps distinct x in the domain to distinct y in the codomain, such that f(x) = y.

Put another way, f is injective if f(a) = f(b) implies a = b (or a ≠ b implies f(a) ≠ f(b)), for any a, b in the domain.

An injective function is called an injection, and is also said to be an information-preserving or one-to-one function (however, the latter name is best avoided, since some authors understand it to mean a one-to-one correspondence, i.e. a bijective function).

A function f that is not injective is sometimes called many-to-one. However, this name too is best avoided, since it is sometimes used to mean "single-valued", i.e. each argument is mapped to at most one value.

Examples and counter-examples

More generally, when X and Y are both the real line R, then an injective function f : R → R is one whose graph is never intersected by any horizontal line more than once.

Injections can be undone

Functions with left inverses (often called sections) are always injections. That is to say, for f : X → Y, if there exists a function g : Y → X such that, for every

(f can be undone by g)

then f is injective. Conversely, it is usually assumed that every injection with non-empty domain has a left inverse.

Note that g may not be a complete inverse of f because the composition in the other order, f o g, may not be the identity on Y. In other words, a function that can be undone or "reversed", such as f, is not necessarily invertible (bijective). Injections are "reversible" but not always invertible.

Although it is impossible to reverse a non-injective (and therefore information-losing) function, you can at least obtain a "quasi-inverse" of it, that is a multiple-valued function.

Injections may be made invertible

In fact, to turn an injective function f : X → Y into a bijective (hence invertible) function, it suffices to replace its codomain Y by its actual range J = f(X). That is, let g : X → J such that g(x) = f(x) for all x in X; then g is bijective. Indeed, f can be factored as incl_J,Yog, where incl_J,Yis the inclusion function from J into Y.

Other properties

• If f and g are both injective, then f o g is injective.

• If g o f is injective, then f is injective (but g need not be).
• f : X → Y is injective if and only if, given any functions g, h : W → X, whenever f o g = f o h, then g = h.
• If f : X → Y is injective and A is a subset of X, then f ⁻¹(f(A)) = A. Thus, A can be recovered from its image f(A).
• If f : X → Y is injective and A and B are both subsets of X, then f(A ∩ B) = f(A) ∩ f(B).
• Every function h : W → Y can be decomposed as h = f o g for a suitable injection f and surjection g. This decomposition is unique up to isomorphism, and f may be thought of as the inclusion function of the range h(W) of h as a subset of the codomain Y of h.
• If f : X → Y is an injective function, then Y has at least as many elements as X, in the sense of cardinal numbers.
• If both X and Y are finite with the same number of elements, then f : X → Y is injective if and only if f is surjective.
• Every embedding is injective.

Category theory view

In the language of category theory, injective functions are precisely the monomorphisms in the category of sets.

See also

• surjective function
• injective module
• monomorphism
• Horizontal line test
• Injective metric space