mathematics, a surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that, for every element y of the function's codomain...

injective non-surjective function (injection, not a bijection) An injective surjective function (bijection) A non-injective surjective function (surjection...

domain; that is, if the image and the codomain of the function are equal. A surjective function is a surjection. Notationally: ∀ y ∈ Y , ∃ x ∈ X , y =...

codomain and the image of a function are the same set; such a function is called surjective or onto. For any non-surjective function f : X → Y , {\displaystyle...

set X is equivalent to counting injective functions N → X when n = x, and also to counting surjective functions N → X when n = x. Counting multisets of...

{\displaystyle y\in Y} implies that f is surjective. The inverse function f −1 to f can be explicitly described as the function f − 1 ( y ) = ( the unique element ...

element of Y. Functions which satisfy property (3) are said to be "onto Y " and are called surjections (or surjective functions). Functions which satisfy...

element x in the domain X. The identity function on X is clearly an injective function as well as a surjective function (its codomain is also its range), so...

partial functions. A partial function is said to be injective, surjective, or bijective when the function given by the restriction of the partial function to...

thus f − 1 ( y ) = { x } . {\displaystyle f^{-1}(y)=\{x\}.} The function f is surjective (or onto, or is a surjection) if its range f ( X ) {\displaystyle...

composition of one-to-one (injective) functions is always one-to-one. Similarly, the composition of onto (surjective) functions is always onto. It follows that...

analogues of onto or surjective functions (and in the category of sets the concept corresponds exactly to the surjective functions), but they may not exactly...

ideal is called a place of K/k. A discrete valuation of K/k is a surjective function v : K → Z∪{∞} such that v(x) = ∞ iff x = 0, v(xy) = v(x) + v(y) and...

Riemann-integrable. The Peano space-filling curve is a continuous surjective function that maps the unit interval [ 0 , 1 ] {\displaystyle [0,1]} onto...

injective function from S {\displaystyle S} to N {\displaystyle \mathbb {N} } . S {\displaystyle S} is empty or there exists a surjective function from N...

elements of a set J, then J is an index set. The indexing consists of a surjective function from J onto A, and the indexed collection is typically called an...

example, to say that a function is onto (surjective) or not the codomain should be taken into account. The graph of a function on its own does not determine...

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept...

one-to-one function. In other words, every element of the function's codomain is the image of at most one element of its domain. Surjective function: has a...

simple-to-understand function which takes on every real value in every interval, that is, it is an everywhere surjective function. It is thus discontinuous...

f. The image of a function is a subset of its codomain so it might not coincide with it. Namely, a function that is not surjective has elements y in its...

In mathematics, the restriction of a function f {\displaystyle f} is a new function, denoted f | A {\displaystyle f\vert _{A}} or f ↾ A , {\displaystyle...

elements. An injective function is called an injection, a surjective function is called a surjection, and a bijective function is called a bijection or...

this equivalence. Any injective function between two finite sets of the same cardinality is also a surjective function (a surjection). Similarly, any surjection...

of X not included in it. That is, X is nonempty and there is no surjective function from the natural numbers to X. The cardinality of X is neither finite...

In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is...

is surjective, this topology is canonically identified with the quotient topology under the equivalence relation defined by f. Dually, for a function f...

{\displaystyle \left(a_{1},\ldots ,a_{n}\right)} may be identified with the (surjective) function F   :   { 1 , … , n }   →   { a 1 , … , a n } {\displaystyle F~:~\left\{1...

cardinality of S is less than the cardinality of T, then there is no surjective function from S to T. Let q1, q2, ..., qn be positive integers. If q 1 + q...

{\displaystyle B} ⁠, that is, a function from ⁠ A {\displaystyle A} ⁠ to ⁠ B {\displaystyle B} ⁠ that is both injective and surjective. Such sets are said to be...