Surjective function

In mathematics, a surjective function (also known as surjection, or onto function y of the function's codomain, there exists Y is the image of the function's domain f may map one or more elements of

A surjective function is a function whose image is equal to its codomain. Equivalently, a function Y is surjective if for every x in U+21A0↠RIGHTWARDS TWO HEADED ARROW), as in f is said to be surjective if

• For any set X, the identity function idX on X is surjective.
• The function "}},"i":0}}]}'/>f : Z → {0, 1} defined by f(n) = n mod 2 (that is, even integers are mapped to 0 and odd integers to 1) is surjective.
• The function g : R → R defined by 2</sup>",{"template":{"target":{"wt":"Nowrap end","href":"./Template:Nowrap_end"},"params":{},"i":1}}]}'>g(x) = x is not surjective, since there is no real number x such that 2</sup> = −1",{"template":{"target":{"wt":"Nowrap end","href":"./Template:Nowrap_end"},"params":{},"i":1}}]}'>x = −1. However, the function x = y.
• The natural logarithm function (0, +∞)</nowiki> → '''R'''"}},"i":0}}]}'/>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, if defined with the set of real numbers as the domain and the codomain, is not surjective (as its range is the set of positive 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 (that is, the group of all n×n invertible matrices). Under this definition, the matrix exponential is surjective for complex matrices, although still not surjective for real matrices.
• The projection from a cartesian product g : Y → X is said to be a right inverse of the function f o g of g and f in that order is the identity function on the domain Y of g. The function g need not be a complete inverse of f because the composition in the other order, o</small> ''f''"}},"i":0}}]}'>g o f, may not be the identity function on the domain X of f. In other words, f can undo or "reverse" g, but cannot necessarily be reversed by it. For example, ⁠(x,y)⁠ only if f : X → Y is surjective and B is a subset of Y, then −1</sup>(''B'')) = ''B''",{"template":{"target":{"wt":"Nowrap end","href":"./Template:Nowrap_end"},"params":{},"i":1}}]}'>f(f(B)) = B. Thus, B can be recovered from its preimage −1</sup>(''B'')"}},"i":0}}]}'>f(B).

  For example, in the first illustration in the gallery, 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 is not unique (it would also work if g(C) equals 3); it only matters that f "reverses" g.

  Surjections as epimorphisms

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  A function g o f = h o f, then g : Y → X satisfying f(g(y)) = y for all y in Y exists. g is easily seen to be injective, thus the formal definition of |Y| ≤ |X| is satisfied.)

  Specifically, if both X and Y are finite with the same number of elements, then f o g is surjective, then f is surjective (but g, the function applied first, need not be). These properties generalize from surjections in the category of sets to any epimorphisms in any category.

  Any function can be decomposed into a surjection and an injection: For any function g : Y → Z such that o</small> ''f''",{"template":{"target":{"wt":"Nowrap end","href":"./Template:Nowrap_end"},"params":{},"i":1}}]}'>h = g o f. To see this, define Y to be the set of preimages −1</sup>(''z'')"}},"i":0}}]}'>h(z) where z is in A and

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  A non-injective surjective function (surjection, not a bijection)
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  An injective surjective function (bijection)
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  An injective non-surjective function (injection, not a bijection)
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  A non-injective non-surjective function (neither a bijection)