injective function

In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. (This is in contrast to a "many-to-one" function, which may map two distinct input values to the same output value.) Note that the phrase "one-to-one" is, in common usage, easily confused with a bijection. An injection does not necessarily cover all possible outputs (i.e., it is not necessarily surjective). A function f : X → Y is injective if, for every y in the codomain Y, there is at most one x in the domain X with f(x) = y. Put another way, f is injective if, for every x and x' in X, whenever f(x) = f(x'), we must have x = x'.

Bijective (injective and surjective)	Injective, not surjective
Surjective, not injective	Not surjective, not injective

When X and Y are both the real line R, then an injective function f : R → R can be visualized as one whose graph is never intersected by any horizontal line more than once (this is the horizontal line test.)

Examples and counterexamples

Consider the function f : R → R defined by f(x) = 2x + 1. This function is injective, since given arbitrary real numbers x and x', if 2x + 1 = 2x' + 1, then 2x = 2x', so x = x'. On the other hand, the function g : R → R defined by g(x) = x² is not injective, because (for example) g(1) = 1 = g(−1). However, if we define the function h : [0, ∞) → R by the same formula as g, but with the domain restricted to only the nonnegative real numbers, then the function h is injective. This is because, given arbitrary nonnegative real numbers x and x', if x² = x'², then |x| = |x'|, so x = x'.

Properties

A function f : X → Y is injective if and only if X is the empty set or there exists a function g : Y → X such that g o f equals the identity function on X.
By definition, a function is bijective if and only if it is both injective and surjective.
If g o f is injective, then f is injective.
If f and g are both injective, then g o f is injective.
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.

Injective Function

Examples and counterexamples

Properties

See also