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Что (кто) такое surjection - определение
FUNCTION SUCH THAT EVERY ELEMENT HAS A PREIMAGE
Surjective; Onto; Onto function; Surjectivity; Onto (mathematics); Surjective map; Surjection; ↠; Surjective Function; Induced function; Onto mapping
![range]]) of ''f''. This function is '''not''' surjective, because the image does not fill the whole codomain. In other words, ''Y'' is colored in a two-step process: First, for every ''x'' in ''X'', the point ''f''(''x'') is colored yellow; Second, all the rest of the points in ''Y'', that are not yellow, are colored blue. The function ''f'' would be surjective only if there were no blue points.](https://commons.wikimedia.org/wiki/Special:FilePath/Codomain2.SVG?width=200 "range]]) of ''f''. This function is '''not''' surjective, because the image does not fill the whole codomain. In other words, ''Y'' is colored in a two-step process: First, for every ''x'' in ''X'', the point ''f''(''x'') is colored yellow; Second, all the rest of the points in ''Y'', that are not yellow, are colored blue. The function ''f'' would be surjective only if there were no blue points.")
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surjection
<mathematics> A function f : A -> B is surjective or onto or a
surjection if f A = B. I.e. f can return any value in B.
This means that its image is its codomain.
Only surjections have right inverses, f' : B -> A where
f (f' x) = x since if f were not a surjection there would be
elements of B for which f' was not defined.
See also bijection, injection.
(1995-05-27)
onto
onto
also on to
Frequency: The word is one of the 3000 most common words in English.
Note: In addition to the uses shown below, 'onto' is used in phrasal verbs such as 'hold onto' and 'latch onto'.
1.
If something moves or is put onto an object or surface, it is then on that object or surface.
I took my bags inside, lowered myself onto the bed and switched on the TV...
Smear Vaseline on to your baby's skin to prevent soreness.
PREP
2.
You can sometimes use onto to mention the place or area that someone moves into.
The players emerged onto the field...
Alex turned his car on to the Albert Quay and drove along until he found a parking place.
PREP
3.
You can use onto to introduce the place towards which a light or someone's look is directed.
...the metal part of the door onto which the sun had been shining...
...the house with its leafy garden and its view on to Regent's Park.
PREP
4.
You can use onto to introduce a place that you would immediately come to after leaving another place that you have just mentioned, because they are next to each other.
...windows opening onto carved black-wood balconies...
The door opened onto a lighted hallway.
PREP: v PREP n
5.
When you change the position of your body, you use onto to introduce the part your body which is now supporting you.
As he stepped backwards she fell onto her knees, then onto her face...
I willed my eyes to open and heaved myself over on to my back.
PREP
6.
When you get onto a bus, train, or plane, you enter it in order to travel somewhere.
As he got on to the plane, he asked me how I was feeling...
'I'll see you onto the train.'-'Thank you.'
? off
PREP
7.
Onto is used after verbs such as 'hold', 'hang', and 'cling' to indicate what someone is holding firmly or where something is being held firmly.
The reflector is held onto the sides of the spacecraft with a frame...
She was conscious of a second man hanging on to the rail...
PREP
8.
If people who are talking get onto a different subject, they begin talking about it.
Let's get on to more important matters...
So, if we could just move onto something else?
PREP
9.
You can sometimes use onto to indicate that something or someone becomes included as a part of a list or system.
The Macedonian question had failed to get on to the agenda...
The pill itself has changed a lot since it first came onto the market...
Twelve thousand workers will go onto a four-day week at their factory in Birmingham.
PREP
10.
If someone is onto something, they are about to discover something important. (INFORMAL)
He leaned across the table and whispered to me, 'I'm really onto something.'...
Archaeologists knew they were onto something big when they started digging.
PREP: be PREP n
11.
If someone is onto you, they have discovered that you are doing something illegal or wrong. (INFORMAL)
I had told people what he had been doing, so now the police were onto him.
PREP: be PREP n
surjective
Onto
·prep On the top of; upon; on. ·see On to, under On, ·prep
Rigidity (mathematics)
TYPE OF PROPERTY OF A COLLECTION OF MATHEMATICAL OBJECTS WHICH ARE UNIQUELY DETERMINED BY LESS INFORMATION THAN ONE WOULD EXPECT
Rigidity theorem; Rigid surjection
In mathematics, a rigid collection C of mathematical objects (for instance sets or functions) is one in which every c ∈ C is uniquely determined by less information about c than one would expect.
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.
equivalence class
MATHEMATICAL CONCEPT
Quotient set; Equivalence classes; Factor space; Canonical projection; Canonical projection map; Equivalence class representative; Equivalence Class Representative; Equivalence Class Of Y; Class representative (mathematics); Quotient sets; Equivalence set; Representative (mathematics); Canonical surjection
<mathematics> An equivalence class is a subset whose elements
are related to each other by an equivalence relation. The
equivalence classes of a set under some relation form a
partition of that set (i.e. any two are either equal or
disjoint and every element of the set is in some class).
(1996-05-13)
Equivalence class
MATHEMATICAL CONCEPT
Quotient set; Equivalence classes; Factor space; Canonical projection; Canonical projection map; Equivalence class representative; Equivalence Class Representative; Equivalence Class Of Y; Class representative (mathematics); Quotient sets; Equivalence set; Representative (mathematics); Canonical surjection
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent.
Википедия
Surjective function
In mathematics, a surjective function (also known as surjection, or onto function ) is a function f such that every element y can be mapped from element x so that f(x) = y. In other words, every element of the function's codomain is the image of at least one element of its domain. It is not required that x be unique; the function f may map one or more elements of X to the same element of Y.
The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word sur means over or above, and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain.
Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse assuming the axiom of choice, and every function with a right inverse is necessarily a surjection. The composition of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection.
