Compositions of Bijective Functions Are Again Bijective

Properties of mathematical functions

	surjective	not-surjective
injective	bijective	injective-only
non- injective	surjective-only	general

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 office maps elements from its domain to elements in its codomain. Given a role ${\displaystyle f\colon Ten\to Y}$ :

• The function is injective, or one-to-i, if each chemical 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.^[i] Notationally:

${\displaystyle \forall x,x'\in X,f(x)=f(x')\implies x=x',}$

or, equivalently (using logical transposition),

${\displaystyle \forall x,ten'\in X,x\neq 10'\implies f(x)\neq f(x').}$ ^{[2] [3] [4]}

• The role is surjective, or onto, if each element of the codomain is mapped to past 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] Notationally:

${\displaystyle \forall y\in Y,\exists 10\in X{\text{ such that }}y=f(x).}$ ^{[2] [3] [4]}

• The part is bijective (1-to-1 and onto, ane-to-ane correspondence, or invertible) if each element of the codomain is mapped to by exactly one chemical element of the domain. That is, the function is both injective and surjective. A bijective function is also called a bijection.^{[1] [2] [iii] [4]} That is, combining the definitions of injective and surjective,

${\displaystyle \forall y\in Y,\exists !x\in X{\text{ such that }}y=f(x),}$

where ${\displaystyle \exists !ten}$ ways "there exists exactly one x".

• In whatsoever case (for any office), the post-obit holds:

${\displaystyle \forall x\in X,\exists !y\in Y{\text{ such that }}y=f(x).}$

An injective function need not exist surjective (not all elements of the codomain may exist 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.

Injection [edit]

Injective composition: the second function demand not exist injective.

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

The part ${\displaystyle f\colon X\to Y}$ is injective, if for all ${\displaystyle ten,x'\in X}$ , ${\displaystyle f(10)=f(x')\Rightarrow x=10'.}$ ^{[2] [3] [4]}

The following are some facts related to injections:

Surjection [edit]

Surjective limerick: the first function need not exist surjective.

A function is surjective or onto if each chemical element of the codomain is mapped to by at least one element of the domain. In other words, each element of the codomain has non-empty preimage. Equivalently, a office is surjective if its paradigm is equal to its codomain. A surjective function is a surjection.^[1] The formal definition is the post-obit.

The function ${\displaystyle f\colon X\to Y}$ is surjective, if for all ${\displaystyle y\in Y}$ , there is ${\displaystyle ten\in X}$ such that ${\displaystyle f(x)=y.}$ ^{[2] [3] [4]}

The following are some facts related to surjections:

• A part ${\displaystyle f\colon X\to Y}$ is surjective if and merely if information technology is right-invertible, that is, if and only if in that location is a function ${\displaystyle g\colon Y\to Ten}$ such that ${\displaystyle f\circ grand=}$ identity role on ${\displaystyle Y}$ . (This statement is equivalent to the axiom of choice.)
• By collapsing all arguments mapping to a given stock-still image, every surjection induces a bijection from a quotient set of its domain to its codomain. More precisely, the preimages under f of the elements of the image of ${\displaystyle f}$ are the equivalence classes of an equivalence relation on the domain of ${\displaystyle f}$ , such that x and y are equivalent if and just they take the same prototype under ${\displaystyle f}$ . Every bit all elements of any i of these equivalence classes are mapped by ${\displaystyle f}$ on the aforementioned element of the codomain, this induces a bijection betwixt the caliber set by this equivalence relation (the set of the equivalence classes) and the image of ${\displaystyle f}$ (which is its codomain when ${\displaystyle f}$ is surjective). Moreover, f is the limerick of the canonical projection from f to the quotient ready, and the bijection between the quotient set and the codomain of ${\displaystyle f}$ .
• The composition of two surjections is again a surjection, but if ${\displaystyle g\circ f}$ is surjective, then it tin can only exist concluded that ${\displaystyle thousand}$ is surjective (see figure).

Bijection [edit]

Bijective composition: the showtime part demand not be surjective and the second function need not be injective.

A function is bijective if information technology is both injective and surjective. A bijective part is also chosen a bijection or a one-to-i correspondence. A function is bijective if and but if every possible image is mapped to past exactly one argument.^[one] This equivalent condition is formally expressed equally follow.

The function ${\displaystyle f\colon X\to Y}$ is bijective, if for all ${\displaystyle y\in Y}$ , there is a unique ${\displaystyle ten\in 10}$ such that ${\displaystyle f(x)=y.}$ ^{[ii] [3] [4]}

The following are some facts related to bijections:

Cardinality [edit]

Suppose that one wants to define what it means for two sets to "accept the aforementioned number of elements". One way to practise this is to say that 2 sets "have the same number of elements", if and only if all the elements of ane set can be paired with the elements of the other, in such a way that each element is paired with exactly one element. Appropriately, one can ascertain ii sets to "have the same number of elements"—if there is a bijection between them. In which case, the two sets are said to take the aforementioned cardinality.

Likewise, 1 can say that gear up ${\displaystyle Ten}$ "has fewer than or the same number of elements" equally set ${\displaystyle Y}$ , if at that place is an injection from ${\displaystyle X}$ to ${\displaystyle Y}$ ; one can also say that prepare ${\displaystyle X}$ "has fewer than the number of elements" in fix ${\displaystyle Y}$ , if there is an injection from ${\displaystyle X}$ to ${\displaystyle Y}$ , but not a bijection between ${\displaystyle Ten}$ and ${\displaystyle Y}$ .

Examples [edit]

Information technology is important to specify the domain and codomain of each part, since past irresolute these, functions which announced to be the same may take different backdrop.

Injective and surjective (bijective)

The identity function id _X for every non-empty set X, and thus specifically ${\displaystyle \mathbf {R} \to \mathbf {R} :x\mapsto x.}$

${\displaystyle \mathbf {R} ^{+}\to \mathbf {R} ^{+}:x\mapsto ten^{two}}$ , and thus likewise its inverse ${\displaystyle \mathbf {R} ^{+}\to \mathbf {R} ^{+}:x\mapsto {\sqrt {x}}.}$

The exponential role ${\displaystyle \exp \colon \mathbf {R} \to \mathbf {R} ^{+}:x\mapsto \mathrm {e} ^{ten}}$ (that is, the exponential office with its codomain restricted to its prototype), and thus also its inverse the natural logarithm ${\displaystyle \ln \colon \mathbf {R} ^{+}\to \mathbf {R} :x\mapsto \ln {x}.}$

Injective and non-surjective

The exponential function ${\displaystyle \exp \colon \mathbf {R} \to \mathbf {R} :ten\mapsto \mathrm {east} ^{x}.}$

Non-injective and surjective

${\displaystyle \mathbf {R} \to \mathbf {R} :x\mapsto (x-1)x(ten+i)=10^{iii}-10.}$

${\displaystyle \mathbf {R} \to [-1,ane]:10\mapsto \sin(x).}$

Non-injective and non-surjective

${\displaystyle \mathbf {R} \to \mathbf {R} :10\mapsto \sin(x).}$

Properties [edit]

• For every function f , subset X of the domain and subset Y of the codomain, X ⊂ f ⁻¹ (f(X)) and f(f ⁻ⁱ (Y)) ⊂ Y . If f is injective, and so 10 = f ⁻¹ (f(X)), and if f is surjective, so f(f ^−one (Y)) = Y .
• For every office h : 10 → Y , ane can ascertain a surjection H : X → h(X) : x → h(x) and an injection I : h(X) → Y : y → y . It follows that ${\displaystyle h=I\circ H}$ . This decomposition is unique up to isomorphism.

Category theory [edit]

In the category of sets, injections, surjections, and bijections correspond precisely to monomorphisms, epimorphisms, and isomorphisms, respectively.^[5]

History [edit]

The injective-surjective-bijective terminology (both every bit nouns and adjectives) was originally coined by the French Bourbaki grouping, before their widespread adoption.^[6]

Come across also [edit]

• Horizontal line test
• Injective module
• Permutation

References [edit]

1. ^ ^{a b c d e f} "Injective, Surjective and Bijective". www.mathsisfun.com . Retrieved 2019-12-07 .
2. ^ ^{a b c d e f} "Bijection, Injection, And Surjection | Brilliant Math & Scientific discipline Wiki". brilliant.org . Retrieved 2019-12-07 .
3. ^ ^{a b c d e f} Farlow, Due south. J. "Injections, Surjections, and Bijections" (PDF). math.umaine.edu . Retrieved 2019-12-06 .
4. ^ ^{a b c d e f} "6.iii: Injections, Surjections, and Bijections". Mathematics LibreTexts. 2017-09-20. Retrieved 2019-12-07 .
5. ^ "Section 7.3 (00V5): Injective and surjective maps of presheaves—The Stacks project". stacks.math.columbia.edu . Retrieved 2019-12-07 .
6. ^ Mashaal, Maurice (2006). Bourbaki. American Mathematical Soc. p. 106. ISBN978-0-8218-3967-6.

External links [edit]

• Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms.

perezthavy1951.blogspot.com

Source: https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection

Share this post