Definition:
There is a correspondence F between two nonempty sets A and B, and for each element A and B in A, there is always a unique element B corresponding to it. This correspondence is a mapping from A to B, which is marked as F: A → B, where B is called the image of element A under the mapping F, and is marked 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).
In other words, let A and B be two nonempty sets. If any element A in set A has a unique element B corresponding to it according to a certain correspondence F, then the correspondence F: A → B is the mapping from set A to set 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.
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 that there are not necessarily different images in B for different elements in A; 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 corresponding relationship between Set A and Set B, which is also called one-to-one mapping from A to B..
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. )