The notion of preimage is used in set theory.
The notion of preimage is used in the field of mathematics , specifically within the framework of set theory . Before moving forward with the definition of the term, it is advisable to clarify several concepts.
Sets , in this framework, are abstract groupings of elements , such as functions, numbers, letters, etc. The relationship that allows each element of a first set to be assigned an element of a second set, or none, is called a function .
Functions , therefore, are links between the elements of two sets : the starting set (also called the domain ) and the arrival set (called the codomain ).
Preimage concept
With these questions clear, we can define what a preimage is. This is the name given to each element that is part of the starting set . The elements of the arrival set, for their part, are referred to as images .
Preimages, in short, are the elements of the domain . A mathematical function assigns each preimage one image, or none. It is a correspondence that relates the elements of two non-empty sets.
An example
Let us take the case of a starting set formed by the elements “Buenos Aires” , “Montevideo” and “Caracas” , and an arrival set that presents the elements “Argentina” , “Uruguay” and “Venezuela” . Both sets are linked by the function “is the capital of” , which establishes the following relationships: “Buenos Aires” -> “Argentina” ; “Montevideo” -> “Uruguay” ; and “Caracas” -> “Venezuela” .
As you can see, “Buenos Aires” , “Montevideo” and “Caracas” are the pre-images, while “Argentina” , “Uruguay” and “Venezuela” are the images.
If we take a starting set composed of “Buenos Aires”, “Montevideo” and “Caracas”, and an arrival set with “Argentina”, “Uruguay” and “Venezuela”, both related by the function “is the capital of” , we can affirm that “Buenos Aires” is a pre-image.
Range and preimages
In this context, we talk about the concept of range (which is also known by the names scope and range ) 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 the possible values for each of the preimages of the domain. Knowing this potential list can save us a lot of time and work, both in paper and computer research, because it allows us to leave out a number of elements from the arrival set that could never be images of the function.
The application of the route or scope
If we return to the example of country capitals, we can explain clearly and concisely one of the uses of combining the concepts of range and preimage. Since the starting set has the names of the capital cities and the function is intended to relate them to their respective countries, any element in the arriving set that does not meet this requirement is out of range.
In this particular case there are not as many possible restrictions as in relationships between numbers, but we could still establish certain conditions to restrict the results. For example, the function could require that countries only belong to a certain continent if it already knew that property of capitals, to save useless search work on the rest. If all the countries in the world were included in the set of countries but we knew with certainty that the preimages were 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 knowing or not knowing a result is something circumstantial, a portion of reality; Although 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 considered this account and solved it).