Bijection, Injection and Surjection

Get Bijection, Injection and Surjection essential facts below. View Videos or join the Bijection, Injection and Surjection discussion. Add Bijection, Injection and Surjection to your PopFlock.com topic list for future reference or share this resource on social media.

## Injection

## Surjection

## Bijection

### Cardinality

## Examples

## Properties

## Category theory

## History

## See also

## References

## External links

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Bijection, Injection and Surjection

In mathematics, **injections**, **surjections** and **bijections** are classes of functions distinguished by the manner in which *arguments* (input expressions from the domain) and *images* (output expressions from the codomain) are related or *mapped to* each other.

A function maps elements from its domain to elements in its codomain. Given a function :

- The function is
**injective**, or**one-to-one**, if each element of the codomain is mapped to by*at most*one element of the domain, or equivalently, if distinct elements of the domain map to distinct elements in the codomain. An injective function is also called an**injection**.^{[1]}^{[2]}Notationally:

- Or, equivalently (using logical transposition),
^{[3]}^{[4]}^{[5]}

- The function is
**surjective**, or**onto**, if each element of the codomain is mapped to by*at least*one element of the domain. That is, the image and the codomain of the function are equal. A surjective function is a**surjection**.^{[1]}^{[2]}Notationally:

^{[3]}^{[4]}^{[5]}

- The function is
**bijective**(**one-to-one and onto**or**one-to-one correspondence**) if each element of the codomain is mapped to by*exactly*one element of the domain. That is, the function is*both*injective and surjective. A bijective function is also called a**bijection**.^{[1]}^{[2]}^{[3]}^{[4]}^{[5]}

An injective function need not be surjective (not all elements of the codomain may be associated with arguments), and a surjective function need not be injective (some images may be associated with *more than one* argument). The four possible combinations of injective and surjective features are illustrated in the adjacent diagrams.

A function is **injective** (**one-to-one**) if each possible element of the codomain is mapped to by at most one argument. Equivalently, a function is injective if it maps distinct arguments to distinct images. An injective function is an **injection**.^{[1]}^{[2]} The formal definition is the following.

- The function is injective, if for all ,
^{[3]}^{[4]}^{[5]}

The following are some facts related to injections:

- A function
*f*:*X*->*Y*is injective if and only if*X*is empty or*f*is left-invertible; that is, there is a function g : f(X) -> X such that*g*o*f*= identity function on*X*. Here, f(X) is the image of f. - Since every function is surjective when its codomain is restricted to its image, every injection induces a bijection onto its image.
^{[1]}More precisely, every injection*f*:*X*->*Y*can be factored as a bijection followed by an inclusion as follows. Let*f*_{R}:*X*->*f*(*X*) be*f*with codomain restricted to its image, and let*i*:*f*(*X*) ->*Y*be the inclusion map from*f*(*X*) into*Y*. Then*f*=*i*o*f*_{R}. A dual factorisation is given for surjections below. - The composition of two injections is again an injection, but if
*g*o*f*is injective, then it can only be concluded that*f*is injective (see figure). - Every embedding is injective.

A function is **surjective** (**onto**) if each possible image is mapped to by at least one argument. In other words, each element in the codomain has non-empty preimage. Equivalently, a function is surjective if its image is equal to its codomain. A surjective function is a **surjection**.^{[1]}^{[2]} The formal definition is the following.

- The function is surjective, if for all , there is such that
^{[3]}^{[4]}^{[5]}

The following are some facts related to surjections:

- A function
*f*:*X*->*Y*is surjective if and only if it is right-invertible, that is, if and only if there is a function*g*:*Y*->*X*such that*f*o*g*= identity function on*Y*. (This statement is equivalent to the axiom of choice.) - 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*:*X*->*Y*can be factored as a non-bijection followed by a bijection as follows. Let*X*/~ be the equivalence classes of*X*under the following equivalence relation:*x*~*y*if and only if*f*(*x*) =*f*(*y*). Equivalently,*X*/~ is the set of all preimages under*f*. Let*P*(~) :*X*->*X*/~ be the projection map which sends each*x*in*X*to its equivalence class [*x*]_{~}, and let*f*_{P}:*X*/~ ->*Y*be the well-defined function given by*f*_{P}([*x*]_{~}) =*f*(*x*). Then*f*=*f*_{P}o*P*(~). A dual factorisation is given for injections above. - The composition of two surjections is again a surjection, but if
*g*o*f*is surjective, then it can only be concluded that*g*is surjective (see figure).

A function is **bijective** if it is both injective and surjective. A bijective function is a **bijection** (**one-to-one correspondence**).^{[1]} A function is bijective if and only if every possible image is mapped to by exactly one argument.^{[2]} This equivalent condition is formally expressed as follow.

- The function is bijective, if for all , there is a unique such that
^{[3]}^{[4]}^{[5]}

The following are some facts related to bijections:

- A function
*f*:*X*->*Y*is bijective if and only if it is invertible, that is, there is a function*g*:*Y*->*X*such that*g*o*f*= identity function on*X*and*f*o*g*= identity function on*Y*. This function maps each image to its unique preimage. - The composition of two bijections is again a bijection, but if
*g*o*f*is a bijection, then it can only be concluded that*f*is injective and*g*is surjective (see the figure at right and the remarks above regarding injections and surjections). - The bijections from a set to itself form a group under composition, called the symmetric group.

Suppose that one wants to define what it means for two sets to "have the same number of elements". One way to do this is to say that two sets "have the same number of elements", if and only if all the elements of one set can be paired with the elements of the other, in such a way that each element is paired with exactly one element. Accordingly, one can define two sets to "have the same number of elements"--if there is a bijection between them. In which case, the two sets are said to have the same cardinality.

Likewise, one can say that set "has fewer than or the same number of elements" as set , if there is an injection from to ; one can also say that set "has fewer than the number of elements" in set , if there is an injection from to , but not a bijection between and .

It is important to specify the domain and codomain of each function, since by changing these, functions which appear to be the same may have different properties.

- Injective and surjective (bijective)
- The identity function id
_{X}for every non-empty set*X*, and thus specifically - , and thus also its inverse
- The exponential function (that is, the exponential function with its codomain restricted to its image), and thus also its inverse the natural logarithm

- Injective and non-surjective
- The exponential function

- Non-injective and surjective

- Non-injective and non-surjective

- For every function
*f*, subset*X*of the domain and subset*Y*of the codomain,*X*?*f*^{-1}(*f*(*X*)) and*f*(*f*^{-1}(*Y*)) ?*Y*. If*f*is injective, then*X*=*f*^{-1}(*f*(*X*)), and if*f*is surjective, then*f*(*f*^{-1}(*Y*)) =*Y*. - For every function
*h*:*X*->*Y*, one can define a surjection*H*:*X*->*h*(*X*) :*x*->*h*(*x*) and an injection*I*:*h*(*X*) ->*Y*:*y*->*y*. It follows that . This decomposition is unique up to isomorphism.

In the category of sets, injections, surjections, and bijections correspond precisely to monomorphisms, epimorphisms, and isomorphisms, respectively.^{[6]}

The injective-surjective-bijective terminology (both as nouns and adjectives) was originally coined by the French Bourbaki group, before their widespread adoption.^{[7]}

- Bijective function
- Horizontal line test
- Injective module
- Injective function
- Permutation
- Surjective function

- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}"The Definitive Glossary of Higher Mathematical Jargon".*Math Vault*. 2019-08-01. Retrieved . - ^
^{a}^{b}^{c}^{d}^{e}^{f}"Injective, Surjective and Bijective".*www.mathsisfun.com*. Retrieved . - ^
^{a}^{b}^{c}^{d}^{e}^{f}"Bijection, Injection, And Surjection | Brilliant Math & Science Wiki".*brilliant.org*. Retrieved . - ^
^{a}^{b}^{c}^{d}^{e}^{f}Farlow, S. J. "Injections, Surjections, and Bijections" (PDF).*math.umaine.edu*. Retrieved . - ^
^{a}^{b}^{c}^{d}^{e}^{f}"6.3: Injections, Surjections, and Bijections".*Mathematics LibreTexts*. 2017-09-20. Retrieved . **^**"Section 7.3 (00V5): Injective and surjective maps of presheaves--The Stacks project".*stacks.math.columbia.edu*. Retrieved .**^**Mashaal, Maurice (2006).*Bourbaki*. American Mathematical Soc. p. 106. ISBN 978-0-8218-3967-6.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Popular Products

Music Scenes

Popular Artists