In mathematics, an injective function is a function that associates distinct arguments in one domain with distinct values in some codomain; in other words, every unique argument produces a unique result. If all of the elements in the codomain are also mapped, the function is said to be bijective (see figures).

An injective function is called an injection, and is also said to be a one-to-one function (not to be confused with one-to-one correspondence, i.e. a bijective function).

A function f that is not injective is sometimes called many-to-one. (However, this terminology is also sometimes used to mean "single-valued", i.e. each argument is mapped to at most one value.)

A monomorphism is a generalization of an injective function in category theory.

Definition

Let f be a function whose domain is a set A. The function f is injective if for all a and b in A, if f(a) = f(b), then a = b; that is, f(a) = f(b) implies a = b.  Equivalently, if a ≠ b, then f(a) ≠ f(b).

Examples

• For any set X, the identity function on X is injective.

• The function f : R → R defined by f(x) = 2x + 1 is injective.

• The function g : R → R defined by g(x) = x² is not injective, because (for example) g(1) = 1 = g(−1). However, if g is redefined so that its domain is the non-negative real numbers [0,+∞), then g is injective.

• The exponential function exp : R → R defined by exp(x) = e^x is injective (but not surjective as no value maps to a negative number).

• The natural logarithm function ln : (0, ∞) → R defined by x ↦ ln x is injective.

• The function g : R → R defined by g(x) = xⁿ − x is not injective, since, for example, g(0) = g(1).

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. This principle is referred to as the horizontal line test.

Injections can be undone

Functions with left inverses are always injections. That is, given f : X → Y, if there is a function g : Y → X such that, for every x ∈ X


then f is injective. In this case, f is called a section of g and g is called a retraction of f.

Conversely, every injection f with non-empty domain has a left inverse g (in conventional mathematics). Note that g may not be a complete inverse of f because the composition in the other order, f ∘ 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,Y ∘ g, where incl_J,Y is the inclusion function from J into Y.

Other properties

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

• If g ∘ 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 ∘ g = f ∘ h, then g = h. In other words, injective functions are precisely the monomorphisms in the category Set of sets.

• 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 ∘ 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. In particular, if, in addition, there is an injection from $Y$ to $X$, then $X$ and $Y$ has the same cardinal number. (This is known as the Cantor–Bernstein–Schroeder theorem.)

• 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.

• An injective function which is a homomorphism between two algebraic structures is an embedding.

See also

• Surjective function

• Injective module

• Horizontal line test

• Injective metric space