# 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** .

## What is the bijective function?

The bijective function is a function that is both **injective** and **surjective** , which means that if all the elements of the final product Y have a single element of the initial set X to which the condition of the surjective function corresponds.

- Definition
- Properties of the bijective function
- Examples of the bijective function

## 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.