In the surjective function, each element of the second set is constituted as the image of one or more elements of the first set.
In the field of mathematics , a function is called the relationship established between two sets through which, to each of the elements of the first set , an element - or none - of the second is assigned . Depending on their characteristics, there are different types of functions, such as the injective function , the logarithmic function , the exponential function and the quadratic function , among many others.
Overview
The surjective function implies that each element of the second set is the image of at least one element of the first set. This function is also known as subjective , surjective , suprajective , epijective or exhaustive .
We must take into account some essential concepts:
* domain : it is the set that is usually associated with the variable x , that is, with the "input" set. Note that this term refers to "the values that can enter";
* codomain : On the other side of the domain, is the output set, which in many cases (except in the case of a surjective function) contains many values that will not be used;
* image : The codomain values that actually come out of a function.
Characteristics of a surjective function
It can be said that, in a surjective function, each element of the second set (which we can call Y ) has at least one element of the first set ( X ) to which it corresponds.
In formal terms, the surjective function is written this way : f(x) = y . In this way, each y of Y corresponds to one or more x of X.
The surjective function assumes that the path of the function is the second set ( Y ). That is why it can be stated that in a surjective function the path and the domain (starting set or definition set) are equal.
Simple function
Having said all this, we can affirm that the surjective function is characterized, among other things, by being easy to understand and solve. It presents us with a direct path between the input set and the arrival set , because we know in advance that the latter will be output only by elements that have at least one link with another of the first.
Although the complexity of the equation itself can be low or very high, it is not one that can leave us stuck in a step prior to the solution, especially if we analyze it in the opposite direction, from the output set to the entrance one. Mathematics is presented as an exact science, but in its depths there are still shadowy corners , which do not offer a definite answer to every question.
Let us take a case in which the arrival set, the co-domain, contains values that do not relate to any in the domain. This would occur, for example, if the function proposed multiplying by 2 the values assigned to the variable X, requiring that they only be taken from the set of natural numbers , that is, leaving out the decimals.
Given that the contradomain would include all the natural ones and that there is no odd number that results from the multiplication by 2 of another natural one, all of them would be left out of the image , although they continue to belong to the contradomain. In a surjective function, however, there are no rules like the one mentioned above that leave out certain elements from the output set.
The formula of a surjective function is the following: f(x) = y
An example
Let's look at a concrete example to understand what the notion refers to. Let's take the function X → Y defined by f(x) = 4x .
The set X is composed of the elements {2, 4, 6} . The set Y , according to the function, is {8, 16, 24} since
f(2) = 8
f(4) = 16
f(6) = 24
Therefore, f : {2, 4, 6} → {8, 16, 24} defined by f (x) = 4x results in a surjective function .