Surjective function
From Wikipedia, the free encyclopedia
In mathematics, a function is said to be surjective or onto if its range is equal to its codomain. A function f: X → Y is surjective if and only if for every y in the codomain Y there is at least one x in the domain X such that f(x) = y. A surjective function is called a surjection.
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[edit] Examples
For any set X, the identity function idX on X is surjective.
The function f: Z → {0,1,2,3} defined by f(x) = x mod 4 is surjective.
The function f: R → R defined by f(x) = 2x + 1 is surjective (and even bijective), because for every real number y we have an x such that f(x) = y: an appropriate x is (y − 1)/2.
The function g: R → R defined by g(x) = x2 is not surjective, because (for example) there is no real number x such that x2 = −1. However, the function g: R → R+ defined by g(x) = x2 (with restricted codomain) is surjective because for every y in the positive real codomain Y there is at least one x in the real domain X such that x2 = y.
The natural logarithm function ln: (0,+∞) → R is a surjective and even bijective mapping from the set of positive real numbers to the set of all real numbers. Its inverse, the exponential function, is not surjective as its range is the set of positive real numbers and its codomain is usually defined to be the set of all real numbers. The matrix exponential is not surjective when seen as a map from the space of all n×n matrices to itself. It is, however, usually defined as a map from the space of all n×n matrices to the general linear group of degree n, i.e. the group of all n×n invertible matrices. Under this restriction the matrix exponential is surjective.
[edit] There always exists a function "reversible" by a surjection
Every function with a right inverse is a surjection. The converse is equivalent to the axiom of choice. That is, assuming the axiom of choice, a function f: X → Y is surjective if and only if there exists a function g: Y → X such that, for every 
(g can be undone by f)
that is a function g such that f o g equals the identity function on Y (cf. with definition of inverse function).
Note that g may not be a complete inverse of f because the composition in the other order, g o f, may not be the identity on X. In other words, f can undo or "reverse" g, but not necessarily can be reversed by it. Surjections are not always invertible (bijective).
For example, in the first illustration, there is some function g such that g(C) = 4. There is also some function f such that f(4) = C. It doesn't matter that g(C) can also equal 3; it only matters that f "reverses" g.
[edit] Other properties
The composite of surjective functions is always surjective: If f and g are both surjective, then g o f is surjective. Conversely, if f o g is surjective, then f is surjective (but g need not be).
A function f: X → Y is surjective if and only if it is right-cancellative: given any functions g,h:Y → Z, whenever g o f = h o f, then g = h. In other words, surjective functions are precisely the epimorphisms in the category Set of sets.
The cardinality of the domain of a surjective function is greater than or equal to the cardinality of its codomain: If f: X → Y is a surjective function, then X has at least as many elements as Y, in the sense of cardinal numbers. Specifically, if both X and Y are finite with the same number of elements, then f : X → Y is surjective if and only if f is injective.
If f: X → Y is surjective and B is a subset of Y, then f(f −1(B)) = B. Thus, B can be recovered from its preimage f −1(B).
Any function can be decomposed into a surjection and an injection: For any function h: X → Z there exist a surjection f:X → Y and an injection g:Y → Z such that h = g o f. To see this, define Y to be the sets h −1(z) where z is in Z. These sets are disjoint and partition X. Then f carries each x to the element of Y which contains it, and g carries each element of Y to the point in Z to which h sends its points. Then f is surjective since it is a projection map, and g is injective by definition.
By collapsing all arguments mapping to a given fixed image, every surjection induces a bijection defined on a quotient of its domain. More precisely, every surjection f : A → B can be factored as a projection followed by a bijection as follows. Let A/~ be the equivalence classes of A under the following equivalence relation: x ~ y if and only if f(x) = f(y). Equivalently, A/~ is the set of all preimages under f. Let P(~) : A → A/~ be the projection map which sends each x in A to its equivalence class [x]~, and let fP : A/~ → B be the well-defined function given by fP([x]~) = f(x). Then f = fP o P(~).
[edit] Gallery
 A surjective function. (However, this one is not an injection) |
 Another surjective function. (This one happens to be a bijection) |
 A non-surjective function. (This one happens to be an injection) |
 Surjective composition: the first function need not be surjective. |
[edit] See also
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Look up surjective, surjection, or onto in Wiktionary, the free dictionary. |
