Bijective function

The notion of a function is fundamentally important in practically all areas of mathematics , so it is important to learn some basic definitions regarding functions. A bijective function is a function that is both injective and surjective . This function is also known as the one-to-one function , and it is important to make the distinction that you never assign two elements in your domain to the same element in your range .

Definition

We can define or say that a function is bijective if this same function is both  injective and  surjective . This means, if every element that exists in the final set Y has a single element of the initial set X to which it corresponds and what is known as the surjective function condition and all the elements of the initial set X have a single image in the final set Y, which is what we know as an injective function condition .

Another definition that we can give of the term is that a function is bijective if each element of the starting set has a different image in the arrival set , and each element of the arrival set corresponds to an element of the starting set. Theoretically we can say that it is a bijective function if: for all y of Y, there exists a single x of X such that f (x) = y

We must remember at all times that, in a function, whatever, you should always have a set of starting or domain , a set of arrival or against domain , and finally a range .

Properties of the bijective function

Cardinality : When two sets have the same number of elements , at least one way to associate each element of the first set with an element of the second. In this way, there is no excess element in any of the sets. This association that can be made between sets with the same number of elements is called the cardinality of the bijective function .

An association is bijective when each element of the first set corresponds to an element of the second set without there being any excess elements in any of the sets. Two sets have the same cardinal if a bijective function can be established between them.

Bijectivity:  A function f: A → B is bijective and can also be surjective at the same time, that is, each of the elements of set B ( surjectivity ) have to be related to one and only one element of set A ( injectivity ). When there is bijectivity, a rule can be found that reproduces the evolutions of the automaton inversely .

Examples of the bijective function

In a room we find a certain number of seats. A group of spectators enters the room and the exhibitor asks everyone to take a seat. After making a quick observation of the room, the exhibitor declares with certainty that there is a bijectivity between the group of spectators and the number of seats in the place, where each spectator is paired with the seat that corresponds to him. What the teacher had to observe in order to make this statement is:

• All the spectators were sitting in their chairs and no one was standing.
• None of the spectators was sitting in more than one seat.
• Every seat was taken and there were no empty seats.
• No seat was occupied by more than one spectator.

The teacher, through his observation, was able to conclude that there were equal number of seats as there were spectators, without having to count the number of seats.