Injective function
Function  

x ↦ f (x)  
Examples by domain and codomain  


Classes/properties  
Constant · Identity · Linear · Polynomial · Rational · Algebraic · Analytic · Smooth · Continuous · Measurable · Injective · Surjective · Bijective  
Constructions  
Restriction · Composition · λ · Inverse  
Generalizations  
Partial · Multivalued · Implicit  
In mathematics, an injective function or injection or onetoone function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is the image of at most one element of its domain. The term onetoone function must not be confused with onetoone correspondence (a.k.a. bijective function), which uniquely maps all elements in both domain and codomain to each other (see figures).
An injective nonsurjective function (injection, not a bijection)
An injective surjective function (bijection)
A noninjective surjective function (surjection, not a bijection)
A noninjective nonsurjective function (also not a bijection)
Occasionally, an injective function from X to Y is denoted f : X ↣ Y, using an arrow with a barbed tail (U+21A3 ↣ RIGHTWARDS ARROW WITH TAIL).^{[1]} The set of injective functions from X to Y may be denoted Y^{X} using a notation derived from that used for falling factorial powers, since if X and Y are finite sets with respectively m and n elements, the number of injections from X to Y is n^{m} (see the twelvefold way).
A function f that is not injective is sometimes called manytoone. However, the injective terminology is also sometimes used to mean "singlevalued", i.e., each argument is mapped to at most one value.
A monomorphism is a generalization of an injective function in category theory.
Contents
Definition[edit]
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, whenever 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).
Symbolically,
which is logically equivalent to the contrapositive,
Examples[edit]
 For any set X and any subset S of X the inclusion map S → X (which sends any element s of S to itself) is injective. In particular the identity function X → X is always injective (and in fact bijective).
 If the domain X = ∅ or X has only one element, the function X → Y is always injective.
 The function f : R → R defined by f(x) = 2x + 1 is injective.
 The function g : R → R defined by g(x) = x^{2} is not injective, because (for example) g(1) = 1 = g(−1). However, if g is redefined so that its domain is the nonnegative 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 real 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) = x^{n} − x is not injective, since, for example, g(0) = g(1) = 0.
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[edit]
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,
 g(f(x)) = x (f can be undone by g)
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 nonempty domain has a left inverse g, which can be defined by fixing an element a in the domain of f so that g(x) equals the unique preimage of x under f if it exists and g(x) = a otherwise.^{[2]}
The left inverse g is not necessarily an inverse of f because the composition in the other order, f ∘ g, may differ from the identity on Y. In other words, an injective function can be "reversed" by a left inverses, but is not necessarily invertible, which requires that the function is bijective.
Injections may be made invertible[edit]
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.
More generally, injective partial functions are called partial bijections.
Other properties[edit]
 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 have 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 (in which case f is bijective).
 An injective function which is a homomorphism between two algebraic structures is an embedding.
 Unlike surjectivity, which is a relation between the graph of a function and its codomain, injectivity is a property of the graph of the function alone; that is, whether a function f is injective can be decided by only considering the graph (and not the codomain) of f.
Proving that functions are injective[edit]
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 contrapositive of the definition of injectivity, namely that if f(x) = f(y), then x = y.^{[3]}
Here is an example:
 f = 2x + 3
Proof: Let f : X → Y. Suppose f(x) = f(y). So 2x + 3 = 2y + 3 ⇒ 2x = 2y ⇒ 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.
See also[edit]
 Surjective function
 Partial function
 Injective module
 Bijection, injection and surjection
 Injective metric space
Notes[edit]
 ^ "Unicode" (PDF). Retrieved 20130511.
 ^ Unlike the corresponding statement that every surjective function has a right inverse, this does not require the axiom of choice, as the existence of a is implied by the nonemptiness of the domain. However, this statement may fail in less conventional mathematics such as constructive mathematics. In constructive mathematics, the inclusion {0,1} → R of the twoelement set in the reals cannot have a left inverse, as it would violate indecomposability, by giving a retraction of the real line to the set {0,1}.
 ^ Williams, Peter. "Proving Functions OnetoOne". Archived from the original on 4 June 2017.
References[edit]
 Bartle, Robert G. (1976), The Elements of Real Analysis (2nd ed.), New York: John Wiley & Sons, ISBN 9780471054641, p. 17 ff.
 Halmos, Paul R. (1974), Naive Set Theory, New York: Springer, ISBN 9780387900926, p. 38 ff.
External links[edit]
Wikimedia Commons has media related to Injectivity. 
Look up injective in Wiktionary, the free dictionary. 