The concept of mapping What are the differences and similarities between a function and a mapping?

In mathematics, mapping is a term that refers to the relationship between two sets of elements that "correspond" to each other, a noun; it also refers to the action of "forming correspondences", a verb.

The mapping, which requires a predefined function for the projection law part of the operation. Thus "mapping" calculations allow for cross-dimensional correspondences. The corresponding calculus is a purely numerical computation that does not allow for cross-dimensional correspondence, and the use of differential simulation allows for complex simulations in this dimension. Mapping can approximate the correspondence of unrelated sets, whereas calculus can only perform exact operations on a large set of continuously related sets.

Mapping, in mathematics and related fields is often equated with functions. Based on this, a partial mapping is equivalent to a partial function, while a complete mapping is equivalent to a complete function.

In many specific fields of mathematics, the term is used to describe functions that have specific properties associated with that field, e.g., continuous functions in topology, linear transformations in linear algebra, and so on.

The definition of a function is usually divided into the traditional definition and the recent definition; the two definitions of a function are essentially the same, except that they recount the concepts from different starting points; the traditional definition is from the point of view of motion and change, while the recent definition is from the point of view of sets and mappings.

Concepts

In a process of change, the quantity that changes is called a variable (in math, often x, and y changes with the value of x), and there are some values that don't change with the variable, we call them constants.

Autonomous variable (function): a variable that is associated with its quantity, any value in this quantity can find a corresponding fixed value in its quantity.

Dependent variable (function): a variable that changes as the independent variable changes, and when the independent variable takes on a unique value, the dependent variable (function) has and only has a unique value corresponding to it.

function value: in y is a function of x, x determines a value, y determines a value with it, when x takes a, y determines b with it, b is called the function value of a. A function is a special kind of mapping that requires elements of two sets to be numbers, and elements of two sets in a mapping to be arbitrary mathematical objects.

Functions require that each domain of values has a corresponding domain of definitions that corresponds to it, that is, the set of domains of values can have no leftover elements, whereas mappings can have leftovers.