Injective function
Encyclopedia : I : IN : INJ : Injective function
In mathematics, an injective function is a function which associates distinct arguments to distinct values. More precisely, a function f is said to be injective if, for every y in the codomain, there is at most one x in the domain such that f(x) = y.
Put another way, f is injective if f(a) = f(b) implies a = b (or a [\neq] b implies f(a) [\neq] f(b)), for any a, b in the domain.
An injective function is called an injection, and is also said to be information-preserving or, sometimes, one-to-one function. (However, this name is best avoided, since some authors understand it to mean a one-to-one correspondence, i.e. a bijective function.)
A function f that is not injective is sometimes called many-to-one. However, this name too is best avoided, since it is 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 \to \mathbf^+ : x \mapsto \mathrm^x] is injective.
- The natural logarithm function [\ln : (0,+\infty) \to \mathbf : x \mapsto \ln] is injective.
- The function g : R → R defined by [g(x) = x^3 - x ] is not injective, since, for example, g(0) = g(1).
Injections are invertible
Another definition of injection is a function whose effect can be undone. More precisely, f : X → Y is injective if there exists a function g : Y → X such that g(f(x)) = x for every x in ´ X; that is, g o f equals the identity function on X.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 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 inclJ,Yog, where inclJ,Yis the inclusion function from J into Y.
Other properties
- If f and g are both injective, then g o f 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.
- 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.
Category theory view
In the language of category theory, injective functions are precisely the monomorphisms in the category of sets.See also
From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.