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 $f:\; ;\; A\; o\; B$ is injective (one-to-one) if:$forall\; x,\; y\; in\; A,\; f(x)=f(y)\; Rightarrow\; x=y$ or, equivalently, if:$forall\; x,y\; in\; A,\; x\; eq\; y\; Rightarrow\; f(x)\; eq\; f(y).$One could also say that elements of the codomain (sometimes called range) are mapped to by at most one element (argument) of the domain; not every element of the codomain, however, need to have an argument mapped to it. An injective function is an injection.
*A function is surjective (onto) if every element of the codomain is mapped to by some element (argument) of the domain; this is expressed logically by saying that,:$forall\; y\; in\; B,\; exists\; x\; in\; A\; ext\{\; such\; that\; \}\; y\; =\; f(x).$Note that with this definition, some images may be mapped to by more than one argument. (Equivalently, a function where the range is equal to the codomain.) A surjective function is a surjection.
*A function is bijective (one-to-one and onto) if and only if (iff) it is "both" injective and surjective. (Equivalently, "every" element of the codomain is mapped to by "exactly one" element of the domain.) A bijective function is a bijection (one-to-one correspondence).

("Note: a one-to-one function is injective, but may fail to be surjective, while a one-to-one correspondence is both injective and surjective.")

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 following diagrams.

Injection

A function is injective (one-to-one) if every 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. The formal definition is the following.

:The function $f:\; A\; o\; B$ is injective iff for all $a,b\; in\; A$, we have $f(a)\; =\; f(b)\; Rarr\; a\; =\; b.$

*A function "f" : "A" → "B" is injective if and only if "A" is empty or "f" is left-invertible, that is, there is a function "g": "B" → "A" such that "g" o "f" = identity function on "A".
*Since every function is surjective when its codomain is restricted to its range, every injection induces a bijection onto its range. More precisely, every injection "f" : "A" → "B" can be factored as a bijection followed by an inclusion as follows. Let "f"_"R" : "A" → "f"("A") be "f" with codomain restricted to its image, and let "i" : "f"("A") → "B" be the inclusion map from "f"("A") into "B". 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 the figure at right.
*Every embedding is injective.

urjection

A function is surjective (onto) if every possible image is mapped to by at least one argument. In other words, every element in the codomain has non-empty preimage. Equivalently, a function is surjective if its range is equal to its codomain. A surjective function is a surjection. The formal definition is the following.

:The function $f:\; A\; o\; B$ is surjective iff for all $b\; in\; B$, there is $a\; in\; A$ such that $f(a)\; =\; b.$

*A function "f" : "A" → "B" is surjective if and only if it is right-invertible, that is, if and only if there is a function "g": "B" → "A" such that "f" o "g" = identity function on "B". (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" : "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"(~). 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 the figure at right*.

Bijection

A function is bijective if it is both injective and surjective. A bijective function is a bijection (one-to-one correspondence). A function is bijective if and only if every possible image is mapped to by exactly one argument. This equivalent condition is formally expressed as follows.

:The function $f:\; A\; o\; B$ is bijective iff for all $b\; in\; B$, there is a unique $a\; in\; A$ such that $f(a)\; =\; b.$

*A function "f" : "A" → "B" is bijective if and only if it is invertible, that is, there is a function "g": "B" → "A" such that "g" o "f" = identity function on "A" and "f" o "g" = identity function on "B". 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.

Cardinality

Suppose you want 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, we can define two sets to "have the same number of elements" if there is a bijection between them. We say that the two sets have the same cardinality.

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

Examples

It is important to specify the domain and codomain of each function since by changing these, functions which we think of as the same may have different "jectivity".

Injective and surjective (bijective)

* For every set "A" the identity function id_"A" and thus specifically $mathbf\{R\}\; o\; mathbf\{R\}\; :\; x\; mapsto\; x$.
* $mathbf\{R\}^+\; o\; mathbf\{R\}^+\; :\; x\; mapsto\; x^2$ and thus also its inverse $mathbf\{R\}^+\; o\; mathbf\{R\}^+\; :\; x\; mapsto\; sqrt\{x\}$.
* The exponential function $exp\; :\; mathbf\{R\}\; o\; mathbf\{R\}^+\; :\; x\; mapsto\; mathrm\{e\}^x$ and thus also its inverse the natural logarithm $ln\; :\; mathbf\{R\}^+\; o\; mathbf\{R\}\; :\; x\; mapsto\; ln\{x\}$

Injective and non-surjective

* The exponential function $exp\; :\; mathbf\{R\}\; o\; mathbf\{R\}\; :\; x\; mapsto\; mathrm\{e\}^x$

Non-injective and surjective

* $mathbf\{R\}\; o\; mathbf\{R\}\; :\; x\; mapsto\; (x-1)x(x+1)\; =\; x^3\; -\; x$
* $mathbf\{R\}\; o\; [-1,1]\; :\; x\; mapsto\; sin(x)$

Non-injective and non-surjective

* $mathbf\{R\}\; o\; mathbf\{R\}\; :\; x\; mapsto\; x^2$

Properties

* For every function "f", subset "A" of the domain and subset "B" of the codomain we have "A" ⊂ "f" ⁻¹("fA") and "f"("f" ⁻¹"B") ⊂ "B". If "f" is injective we have "A" = "f" ⁻¹("fA") and if "f" is surjective we have "f"("f" ⁻¹"B") = "B".
* For every function "h" : "A" → "C" we can define a surjection "H" : "A" → "h(A)" : a → h(a) and an injection "I" : "h(A)" → "C" : a → a. It follows that "h" = "I" o "H". This decomposition is unique up to isomorphism.

Category theory

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

History

This terminology was originally coined by the Bourbaki group.

ee also

*Injective module
*Permutation
*Horizontal line test