Surjective function

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In mathematics, a function f is said to be surjective or onto, if its values span its whole codomain; that is, for every y in the codomain, there is at least one x in the domain such that f(x) = y .

Said another way, a function f: X → Y is surjective if and only if its range f(X) is equal to its codomain Y. A surjective function is called a surjection.

Contents 1 Examples and a counterexample 2 There always exists a function "reversible" by a surjection 3 Other properties 4 See also

[edit] Examples and a counterexample

• For any set X, the identity function id_X on X 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 natural logarithm function ln: (0,+∞) → R is surjective.
• The function f: Z → {0,1,2,3} defined by f(x) = x mod 4 is surjective.
• 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 → [0,+∞) defined by g(x) = x² (with restricted codomain) 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

• If f and g are both surjective, then f o g is surjective.
• If f o g is surjective, then f is surjective (but g need not be).
• f: X → Y is surjective if and only if, 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.
• 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).
• For any function h: X → Z there exists a surjection f:X → Y and 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 f_P : A/~ → B be the well-defined function given by f_P([x]_~) = f(x). Then f = f_P o P(~).
• If f: X → Y is a surjective function, then X has at least as many elements as Y, in the sense of cardinal numbers.
• 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.

[edit] See also

• Bijective function
• Injective function