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...

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...

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

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

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...

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...

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...

{\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 ...

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...

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

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...

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...

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

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

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...

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...

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...

well-order. Since the collection of all ordinals such that there exists a surjective function from B {\displaystyle B} to the ordinal is a set, there exists an...

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

In category theory, a point-surjective morphism is a morphism f : X → Y {\displaystyle f:X\rightarrow Y} that "behaves" like surjections on the category...

objects. For example, every function may be factored into the composition of a surjective function with an injective function. Matrices possess many kinds...

{\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...

characteristic function of a subset A of some set X maps elements of X to the codomain { 0 , 1 } . {\displaystyle \{0,\,1\}.} This mapping is surjective only when...

element of the frame.) The resulting locale is known as the "locale of surjective functions N → R {\displaystyle \mathbb {N} \to \mathbb {R} } ". The relations...

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...

rank function. Thus the constant rank theorem applies to a generic point of the domain. When the derivative of F is injective (resp. surjective) at a...

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...

of surjective functions f {\displaystyle f} and g {\displaystyle g} also implies the existence of a bijection. We construct an injective function h :...