Injective function

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In mathematics, an injective function is a function which associates distinct arguments with 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 (the latter is 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.)

Examples and counter-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 $exp : mathbf{R} to mathbf{R} : x mapsto mathrm{e}^x$ is injective (but not surjective as no value maps to a negative number).
The natural logarithm function $ln : (0,+infty) to mathbf{R} : x mapsto ln{x}$ is injective.
The function g : R → R defined by $g(x) = x^n - x$ 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.

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 $x in X$

g(f(x)) = x ,

(f can be undone by g)

then f is injective. Conversely, 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. 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 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.

Injective function

Examples and counter-examples

Injections can be undone

Injections may be made invertible

Other properties

See also