 Injective function

"Injective" redirects here. For injective modules, see Injective module.
In mathematics, an injective 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 mapped to by at most one element of its domain. If in addition all of the elements in the codomain are in fact mapped to by some element of the domain, then the function is said to be bijective (see figures).
An injective function is called an injection, and is also said to be a onetoone function (not to be confused with onetoone correspondence, i.e. a bijective function). Occasionally, an injective function from X to Y is denoted f: X ↣ Y, using an arrow with a barbed tail. 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, this terminology is also sometimes used to mean "singlevalued", i.e., each argument is mapped to at most one value; this is the case for any function, but is used to stress the opposition with multivalued functions, which are not true functions.)
A monomorphism is a generalization of an injective function in category theory.
Contents
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 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).
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
 g(f(x)) = x (f can be undone by g)
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 nonempty domain has a left inverse g (in conventional mathematics^{[1]}). 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 noninjective (and therefore informationlosing) function, one can at least obtain a "quasiinverse" of it, that is a multiplevalued 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 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.
See also
 Surjective function
 Bijective function
 Injective module
 Bijection, injection and surjection
 Horizontal line test
 Injective metric space
Notes
 ^ This principle is valid in conventional mathematics, but may fail in constructive mathematics. For instance, a left inverse of the inclusion {0,1} → R of the twoelement set in the reals violates indecomposability by giving a retraction of the real line to the set {0,1}.
References
 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
Categories: Functions and mappings
 Basic concepts in set theory
 Types of functions
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