Mapping means that each element between two sets has a unique corresponding element. That is, if an element has a corresponding element in the first set, then this corresponding element can only correspond to this element in the second set.
Surjectivity means that every element in the first set has a corresponding element in the second set, that is, every element in the second set has at least one corresponding element in the first set.
Injectivity means that every element in the first set has a unique corresponding element in the second set. In other words, each element in the second group has at most one corresponding element in the first group.
In practical application, three corresponding relationships of mapping are widely used. For example, in computer science, one-to-one mapping can be used for data encryption and decryption, full projection can be used for data transmission in network communication, and full projection can be used for data query and update in database.
The simple expression of the mapping condition is:
1, ergodicity of 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.
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).
Note: (1) For different elements in A, there are not necessarily different images in B; (2) Every element in B has an original image (i.e. a surjection), and different elements in Set A have different images (i.e. an injective image) in Set B, then the mapping F establishes a one-to-one correspondence between Set A and Set B, which is also called a one-to-one mapping from A to B..