Among them, b is called the image of element A under the mapping f, and it is recorded as: b = f (a); A is called the original image of B about the mapping F, and the set of images of all elements in the set A is called the range of the mapping F, which is denoted as f(A).
Note: (1) For different elements in A, there are not necessarily different images in B; (2) Every element in B has an original image, and the mapping F establishes a one-to-one correspondence between set A and set B, also known as the one-to-one mapping from A to B. ..
Mapping or projection is also used to define functions in mathematics and related fields. The function is a mapping from a non-empty number set to a non-empty number set, and it can only be a one-to-one mapping or a many-to-one mapping.
In many specific mathematical fields, this term is used to describe functions with specific properties associated with this field, such as continuous functions in topology, linear transformations in linear algebra, and so on.
If we extend the two sets in the function definition from non-empty sets to sets of arbitrary elements (not limited to numbers), we can get the concept of mapping:
Mapping is a mathematical term that describes the special correspondence between two set elements.
According to the definition of mapping, the following mappings are all mappings. (1) Let A={ 1, 2, 3, 4} and B={3, 5, 7, 9}, and the element X in set A corresponds to the element in set B according to the corresponding relationship, which is the mapping from set A to set B.
(2) Let A=N*, B={0, 1}, and the elements in set A correspond to the elements in set B according to the corresponding relation "the remainder obtained by dividing X by 2", which is the mapping from set A to set B.
(3) Let A={x|x is a triangle} and B = {y | y & gt0}, and the element X in set A corresponds to the element in set B according to the correspondence, which is the mapping from set A to set B. ..
(4) Let A=R and B={ Point is on a straight 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 from set A to set B..
(5) Let A={P|P is a point in a rectangular coordinate system} B = {(x, y)|x∈R, y∈R}. According to the method of establishing plane rectangular coordinate system, the point P in A corresponds to the ordered real number pair (x, y) in B, and this correspondence is the mapping from set A to set B..
Mapping has many names in different fields, and its essence is the same. Such as functions, operators and so on. What needs to be explained here is that a function is a mapping between two data sets, and other mappings are not functions.
One-to-one mapping (bijection) is a special mapping, that is, the only correspondence between two groups of elements, usually one-to-one (one-to-one).
(As can be seen from the definition, the correspondence shown in figure 1 is not a mapping, and the correspondence shown in the other three figures is a mapping. )
In other words, let A and B be two nonempty sets. If any element X in set A has a unique element Y corresponding to it according to a certain correspondence F, then the correspondence f:A→B is a function from set A to set B. The simple expressions of the establishment conditions of the mapping are as follows:
Ergodicity of 1. domain: Every element X in X has a corresponding object in the mapping range.
2. Uniqueness of correspondence: one element in the definition domain can only correspond to one element in the mapping range. According to the mapping results, the mapping is classified differently from the following three perspectives:
1. According to the geometric properties of the results, they are classified as surjective (top) and non-surjective (inside).
2. According to the analytical nature of the results, they are divided into injective (one by one) and non-injective.
3. Consider the geometric and analytical properties: complete injection capacity (one-to-one correspondence). The number of elements in set AB is m, n,
Then, the number of mappings from set a to set b is m times of n.
The difference between function and mapping, full mapping and single mapping
Function is the mapping from number set to number set, and this mapping is "full".
That is, the total mapping f: A→B is a function, in which 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.
A "number set" is a set of numbers, which can be integers, rational numbers, real numbers, complex numbers or part of them, and so on.
"Mapping" is a more extensive mathematical concept than function, and it is a definite corresponding relationship from one set to another. That is to say, if F is the mapping from set A to set B, then for any element 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, and the relationship between elements is b = f(a).
A mapping f: A→B is called "full", that is, for all elements in B, A has an original image.
The definition of function should not be surjective, that is to say, the range should be a subset of b (this definition comes from the teaching of general middle schools. In fact, many math books do not necessarily define that a function is surjective. )
A mapping in which every element in an image set has an original image is called surjection:
That is, any element Y in B is an image in A, then F is called a surjection from A to B, emphasizing that F(A)= B(B can have multiple original images).
Different images of different elements in the original image set are called injective images:
If any two different elements in A are all x 1≠x2, and their images are all f(x 1)≠f(x2), then F is called an injectivity from A to B, emphasizing that f(A) is a subset of B.
Monomorphism and surjectivity can be defined as bijectivity.