In mathematics,mappingIt's atermOf two elementscollectionBetween elements“corresponding”Is a noun.mapping, orProjective projectionIn mathematics and related fields, it is often equivalent to a function.Based on this,Partial mappingIs equivalent toPartial function, andFull mappingamount toPerfect function。
TwoNon empty setRoom A and BThere is corresponding relationship f, and for theevery lastThere is always a unique element in element a and BAn element ofbCorresponding to it, which is the mapping from A to B, recorded as f: A → B.Where, b is called theimage[1], recorded as b=f (a).A is called b with respect to the mapping fOriginal image[1]。The set of images of all elements in set A is called the range of mapping f, which is recorded as f (A).[2]
In other words, let A and B be two non empty sets. If according to a certain correspondence f, there is a unique element b in set B corresponding to any element a in set A, then corresponding f is called A → B, which is a mapping from set A to set B.
mapping, orProjective projectionIt is also used to define functions in mathematics and related fields.Function is derived fromNonempty number setThe mapping to non empty number set can only be one-to-one mapping or many to one mapping.
mappingThere are many names in different fields, and their essence is the same.Such as function,operatorwait.It should be noted here that a function is a mapping between two sets of numbers, and other mappings are not functions.One-to-one mapping(Birefringence）It is a special kind of mapping, that is, the unique correspondence between two set elements. Generally speaking, it is one to one（one-on-one）。
Note: (1) Different elements in A may not have different images in B;(2) Each element in B has a primitive image (i.e., surjection), and different elements in set A have different images (i.e., monomorphism) in set B, then mapping f establishes aOne-to-one correspondenceRelation, also called f, is a one-to-one mapping from A to B.[3]
Conditions of establishment
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The conditions for mapping are simply stated as follows:
1．Define FieldsErgodicity of: each element x in X is mapped in therangeThere are corresponding objects in
2. Correspondence uniqueness: one element in the definition field can only correspond to one element in the mapping value field
The definition field is also called the original image set, and the value field is also called the image set.
Number relation
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The number of elements of set AB is m, n,
Then, the number of mappings from set A to set B is
。
Function and mapping, full mapping and single mapping.
A function is a number set to number set mapping, and this mapping is "full".
That is, the full mapping f: A → B is a function, where the original image set A is called the definition domain of the function, and the image set B is called the value domain of the function.
"Number set" is a set of numbers, which can be integers, rational numbers, real numbers, complex numbers, or part of them.
"Mapping" is a broader mathematical concept than function, which is a certain correspondence between one set and another.That is, if f is a mapping from set A to set B, then for any element a in A, there is a unique element b corresponding to a in set B.We call a the original image and b the image.Writing f: A → B, the element relationship is b=f (a)
A mapping f: A → B is called "full", that is, all elements in B exist in AOriginal image。
In the definition of function, it is not required to be a surjection, that is, the range of values should be Bsubset。(This definition comes from the way in ordinary middle schools. In fact, many math books do not necessarily define functions as surjections.)
Image collectionThe mapping where every element in B has an original image is called surjection: that is, if any element y in B is an image in A, then f is called A to BSurjection, emphasizing that f (A)=B (the original image of B can be multiple)
The mapping of different images of different elements in the original image set is called monomorphism: if any two different elements in A x1 ≠ x2, their image f (x1) ≠ f (x2), then f is called monomorphism from A to B, emphasizing that f (A) is Bsubset。
Single shot and full shot can be jointly determined as one-to-one double shot.
classification
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The different classifications of mapping are based on the mapping results from the following three perspectives:
1. Classification according to the geometric properties of the results:Surjection(up) andNon epimorphismInternal
2. Classification according to the analysis nature of the results:Monomorphism(11) andNon monomorphism
3. Consider both geometric and analytical properties:Full monomorphism（One-to-one correspondence）。
If two functions in the function definitionaggregatefromNon empty setExpand toarbitrarilyThe set of elements (not limited to numbers), we can get the concept of mapping: mapping is a special kind of mathematical description between two set elementscorrespondingA term for relationships.
according tomappingThe following corresponding definitions aremapping。
(1) Let A={1,2,3,4}, B={3,5,7,9},aggregateThe element x in A is multiplied by 2 plus 1 according to the corresponding relationshipelementCorrespondence, this correspondence is the mapping of set A to set B.
(2) Let A=N *, B={0,1}, the elements in set A correspond to the elements in set B according to the corresponding relationship "the remainder obtained by dividing x by 2", which is the mapping of set A to set B.
(3) Let A={x | x be a triangle}, B={y | y>0}, and the element x in set A corresponds to the element in set B according to the corresponding relationship "calculation area", which is the mapping from set A to set B.
(4) Set A=R, B={point on the line}, according to the method of establishing the number axis, the number x in A corresponds to the point P in B, which is the mapping of set A to set B.
(5) Let A={P | P be a point} in a rectangular coordinate system, B={(x, y) | x ∈ R, y ∈ R}, according to the method of establishing a plane rectangular coordinate system, it is the point P in A and the point P in BOrdered real numberFor (x, y) correspondence, this correspondence is the mapping of set A to set B.
For "mapping" or "projection", you need to define the function of the projection rule part in advance before performing the operation.Therefore, "mapping" calculation can realize cross dimension correspondence.The corresponding calculus belongs to pure digital calculation, which cannot achieve cross dimensional correspondence. The use of differential simulation can achieve complex simulation within this dimension.Mapping can make corresponding approximation to multiple sets that are not relatedoperationWhile calculus can only be performed in a large set of continuous correlationsaccurateOperation.
