Mapping is a mathematical term that refers to the relationship between elements of two sets. It is often equivalent to a function in mathematics and related fields. Based on this, partial mapping is equivalent to partial function, and complete mapping is equivalent to complete function.

First, the concept:

1, 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.

2. Mapping has many names in different fields, and the 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. Mapping (bijection) is a special mapping, that is, the only correspondence between two groups of elements, which is one-to-one (one-to-one) in popular parlance.

Second, the establishment conditions:

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.

Third, the classification:

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, they are classified as injective (one by one) and non-injective.

3. Consider the geometric and analytical properties: complete injection capacity (one-to-one correspondence).