The notion of preimage It is used in the field of math, specifically within the framework of set theory. Before proceeding with the definition of the term, it is convenient to clarify several concepts.
The sets, in this framework, are abstract groupings of elements, such as functions, numbers, letters, etc. The relation that allows to assign to each element of a first set an element of a second set, or none, is called function.
The functions, therefore, they are links between the elements of two sets: the starting set (also called domain) and the arrival set (called codomain).
With these questions clear, we can define what is a preimage. This is how it is called each element that is part of the starting set. The elements of the arrival set, for their part, are mentioned as images.
The pre-images, in short, are the domain elements. A mathematical function assigns each preimage one image, or none at all. It is a correspondence that relates to the elements of two nonempty sets.
Let us take the case of a starting set formed by the elements “Buenos Aires”, “Montevideo” and “Caracas”, and of an arrival set that presents the elements “Argentina”, “Uruguay” and “Venezuela”. Both sets are linked by the function “It is the capital of”, which establishes the following relationships: “Buenos Aires” -> “Argentina”; “Montevideo” -> “Uruguay”; and “Caracas” -> “Venezuela”. As you can appreciate, “Buenos Aires”, “Montevideo” and “Caracas” are the pre-images, while “Argentina”, “Uruguay” and “Venezuela” are the images.
In this context, the concept of rank (also known by the names of ambit and route) to refer to the set of images of a given function. It is, in other words, a subset of the codomain. It is possible to represent the range as f R or f A.
The importance of the range is very great since it is about the values possible for each of the domain’s preimages. Knowing this potential list can save us a lot of time and work, both in paper and computerized investigations, because it allows us to leave out a number of elements of the arrival set that could never be images of the function.
If we return to the example of country capitals, we can clearly and concisely explain one of the benefits of combining the concepts of rank and preimage. Since the starting set has the names of the capitals and that the function intends to relate them to their respective countries, any element of the arrival set that does not meet this requirement is out of range.
In this particular case there are not so many restrictions possible as in relations between numbers, but anyway we could establish certain conditions to restrict the results. For example, the function could require that countries belong only to a certain continent if it already knew that property of capitals, to save useless search work in the rest. If all the countries in the world were to be found in the group but we knew with certainty that the pre-images are located in America, then we could leave out Asia, Europe, Oceania, Antarctica and Africa.
All this shows us that the preimage has such a character that it determines the potential images, whether the person trying to solve the given equation knows it or not. We must not forget that the knowledge or not of a result is something circumstantial, a portion of reality; Even if no one has applied a certain function to a problem, the resulting values have always existed (1 + 1 has always been 2, even before someone asked this bill and solve it).