surjective

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In , a surjective function (also known as surjection, or onto function) is a that maps an element to every element ; that is, for every , there is an such that . In other words, every element of the function's is the of one element of its . It is not required that be ; the function may map one or more elements of to the same element of . The term ''surjective'' and the related terms ' and ' were introduced by , a group of mainly 20th-century s who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word ' means ''over'' or ''above'', and relates to the fact that the of the domain of a surjective function completely covers the function's codomain. Any function induces a surjection by its codomain to the image of its domain. Every surjective function has a , and every function with a right inverse is necessarily a surjection. The of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection.

# Definition

A surjective function is a whose is equal to its . Equivalently, a function $f$ with $X$ and codomain $Y$ is surjective if for every $y$ in $Y$ there exists at least one $x$ in $X$ with $f\left(x\right)=y$. Surjections are sometimes denoted by a two-headed rightwards arrow (), as in $f\colon X\twoheadrightarrow Y$. Symbolically, :If $f\colon X \rightarrow Y$, then $f$ is said to be surjective if :$\forall y \in Y, \, \exists x \in X, \;\; f\left(x\right)=y$.

# Examples

* For any set ''X'', the id''X'' on ''X'' is surjective. * The function defined by ''f''(''n'') = ''n'' 2 (that is, s are mapped to 0 and integers to 1) is surjective. * The function defined by ''f''(''x'') = 2''x'' + 1 is surjective (and even ), because for every ''y'', we have an ''x'' such that ''f''(''x'') = ''y'': such an appropriate ''x'' is (''y'' − 1)/2. * The function defined by ''f''(''x'') = ''x''3 − 3''x'' is surjective, because the pre-image of any ''y'' is the solution set of the cubic polynomial equation ''x''3 − 3''x'' − ''y'' = 0, and every cubic polynomial with real coefficients has at least one real root. However, this function is not (and hence not ), since, for example, the pre-image of ''y'' = 2 is . (In fact, the pre-image of this function for every ''y'', −2 ≤ ''y'' ≤ 2 has more than one element.) * The function defined by ''g''(''x'') = ''x''2 is ''not'' surjective, since there is no real number ''x'' such that ''x''2 = −1. However, the function defined by (with the restricted codomain) ''is'' surjective, since for every ''y'' in the nonnegative real codomain ''Y'', there is at least one ''x'' in the real domain ''X'' such that ''x''2 = ''y''. * The function is a surjective and even bijective (mapping from the set of positive real numbers to the set of all real numbers). Its inverse, the , if defined with the set of real numbers as the domain, is not surjective (as its range is the set of positive real numbers). * The is not surjective when seen as a map from the space of all ''n''×''n'' to itself. It is, however, usually defined as a map from the space of all ''n''×''n'' matrices to the of degree ''n'' (that is, the of all ''n''×''n'' ). Under this definition, the matrix exponential is surjective for complex matrices, although still not surjective for real matrices. * The from a to one of its factors is surjective, unless the other factor is empty. * In a 3D video game, vectors are projected onto a 2D flat screen by means of a surjective function.

# Properties

A function is if and only if it is both surjective and . If (as is often done) a function is identified with its , then surjectivity is not a property of the function itself, but rather a property of the . This is, the function together with its codomain. Unlike injectivity, surjectivity cannot be read off of the graph of the function alone.

## Surjections as right invertible functions

The function is said to be a of the function if ''f''(''g''(''y'')) = ''y'' for every ''y'' in ''Y'' (''g'' can be undone by ''f''). In other words, ''g'' is a right inverse of ''f'' if the of ''g'' and ''f'' in that order is the on the domain ''Y'' of ''g''. The function ''g'' need not be a complete of ''f'' because the composition in the other order, , 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. Every function with a right inverse is necessarily a surjection. The proposition that every surjective function has a right inverse is equivalent to the . If is surjective and ''B'' is a of ''Y'', then ''f''(''f'' −1(''B'')) = ''B''. Thus, ''B'' can be recovered from its . For example, in the first illustration above, 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''. File:Surjective composition.svg, Surjective composition: the first function need not be surjective. File:Bijection.svg, Another surjective function. (This one happens to be a ) File:Injection.svg, A non-surjective function. (This one happens to be an )

## Surjections as epimorphisms

A function is surjective if and only if it is : given any functions , whenever ''g'' o ''f'' = ''h'' o ''f'', then ''g'' = ''h''. This property is formulated in terms of functions and their and can be generalized to the more general notion of the s of a and their composition. Right-cancellative morphisms are called s. Specifically, surjective functions are precisely the epimorphisms in the . The prefix ''epi'' is derived from the Greek preposition ''ἐπί'' meaning ''over'', ''above'', ''on''. Any morphism with a right inverse is an epimorphism, but the converse is not true in general. A right inverse ''g'' of a morphism ''f'' is called a of ''f''. A morphism with a right inverse is called a .

## Surjections as binary relations

Any function with domain ''X'' and codomain ''Y'' can be seen as a and binary relation between ''X'' and ''Y'' by identifying it with its . A surjective function with domain ''X'' and codomain ''Y'' is then a binary relation between ''X'' and ''Y'' that is right-unique and both left-total and .

## Cardinality of the domain of a surjection

The of the domain of a surjective function is greater than or equal to the cardinality of its codomain: If is a surjective function, then ''X'' has at least as many elements as ''Y'', in the sense of s. (The proof appeals to the to show that a function satisfying ''f''(''g''(''y'')) = ''y'' for all ''y'' in ''Y'' exists. ''g'' is easily seen to be injective, thus the of , ''Y'', ≤ , ''X'', is satisfied.) Specifically, if both ''X'' and ''Y'' are with the same number of elements, then is surjective if and only if ''f'' is . Given two sets ''X'' and ''Y'', the notation is used to say that either ''X'' is empty or that there is a surjection from ''Y'' onto ''X''. Using the axiom of choice one can show that and together imply that , ''Y'', = , ''X'', , a variant of the .

## Composition and decomposition

The of surjective functions is always surjective: If ''f'' and ''g'' are both surjective, and the codomain of ''g'' is equal to the domain of ''f'', then is surjective. Conversely, if is surjective, then ''f'' is surjective (but ''g'', the function applied first, need not be). These properties generalize from surjections in the to any s in any . Any function can be decomposed into a surjection and an : For any function there exist a surjection and an injection such that ''h'' = ''g'' o ''f''. To see this, define ''Y'' to be the set of s where ''z'' is in . These preimages are and ''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.

## Induced surjection and induced bijection

Any function induces a surjection by restricting its codomain to its range. Any surjective function induces a bijection defined on a of its domain by collapsing all arguments mapping to a given fixed image. More precisely, every surjection can be factored as a projection followed by a bijection as follows. Let ''A''/~ be the es of ''A'' under the following : ''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 which sends each ''x'' in ''A'' to its equivalence class 'x''sub>~, and let ''f''''P'' : ''A''/~ → ''B'' be the well-defined function given by ''f''''P''( 'x''sub>~) = ''f''(''x''). Then ''f'' = ''f''''P'' o ''P''(~).

# Space of surjections

Given fixed and , one can form the set of surjections . The of this set is one of the twelve aspects of Rota's , and is given by $, B, !\begin, A, \\, B, \end$, where $\begin, A, \\, B, \end$ denotes a .