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.
In formal logic, this term is sometimes used to represent function predicates, where function is the model of predicates in set theory.
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:
Let a and b be two groups. If there is a unique element in set B that corresponds to any element in set A according to a certain correspondence F, then such correspondence (including sets A and B, and the correspondence F from set A to set B) is called the mapping from set A to set B, which is denoted as F: A → B.
According to the definition of mapping, the following mappings are all mappings.
(1) Let A={ 1, 2, 3, 4} and B={3, 4, 5, 6, 7, 8, 9}, and the element X in the set A corresponds to the element 2x+ 1 in the set B according to the corresponding relationship, corresponding to the sets A to.
⑵ 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..
Given a mapping from set A to set B, and a∈A and b∈B, if element A and element B correspond, then we say that element B is the image of element A, and element A is the original image of element B.
Mapping describes the special correspondence between two set elements in mathematics.
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 (many-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 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 called the mapping from set A to set B.
The simple expressions of the establishment conditions of the mapping are as follows:
1, defining the ergodicity of the domain: each element X in X has a corresponding object in the mapped value domain;
2. Uniqueness of correspondence: one element in the definition domain can only correspond to one element in the mapping range;
Classification of mapping:
Different mapping classification is based on the mapping results, from the following three angles:
1, classified according to the geometric properties of the results: surjective (upper) and non-surjective (inner);
2. According to the analytical nature of the results, it can be divided into injective (one by one) and non-simple;
3. Consider the geometric and analytical properties: complete injection capacity (one-to-one correspondence).