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Injective function

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When we talk about the area of mathematics , especially in the field of functions , it is also very important to know that the function is the link that develops between two sets that are different , a link through which, to each of these elements from one set are assigned only one element from another set or none at all. The idea of an injective function , on the other hand, refers to the property that tells us that two different elements of a first set are concerned with two totally different elements of a second. set that is not equal to the first.

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What is the injective function?

The injective function is the type of function that indicates that the different elements that an initial set or domain has, correspond to different elements of the final set or codomain , and each of these do not have a pre-image of the domain .

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  • Definition of injective function
  • Properties of the injective function
  • Applications
  • Check if it is an injective function
  • Examples

Definition of injective function

The injective function is also known as the one-to-one function . A function can become injective if each of the elements that the final set Y has has a single element of the initial set X to which it corresponds. This means in other words that there cannot be more than one value of X that has the same image Y. Not all the elements of the final set Y must correspond to some element that exists in the initial set X.

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We can say that the injective function occurs when each of the elements that has the domain does not correspond or cannot have more than one image in the codomain . In the area of mathematics , a function f: X   Y is injective if elements that are different from the set X or domain , correspond to different elements in the set Y or codomain of f. This means that each of the elements of the set Y has at most one pre-image in X, or, what is the same, in the set X there cannot be two or more elements that have the same image .

Properties of the injective function

Among the properties of the injective function we mention the following:

  • The notion of correspondence has a basic role in the concept of relationship and function.
  • If n is an odd number , then its domain is the entire set of real numbers .
  • If n turns out to be an even number , the domain will then be formed by the values that make the radicand positive or zero.

Applications

Injective functions serve us or are applied in the correct graphing of the different functions ; if the function of a single real variable is injective, any horizontal line will intersect only at one point. They are also applied to know if the function is invertible . Also to be able to make a classification of linear transformations (injective monomorphism), epimorphism (surjective), isomorphism (bijective).

Check if it is an injective function

In mathematics, a function is injective if given two points a  and  b :

 f (x a ) = f (x b ) ⇒ x a  = x b

For this reason we can say that the function is injective if it manages to fulfill the values of its domain 0  ≠  1 ⇒ f (x 0 ) ≠ f (x 1 ) .  

To check if the function is injective, it can also be done by graphically checking the injectivity of the function , and this is done when any line that is parallel to the X axis cuts the same line, at most, in a point .

Examples

Some examples of the injective function that we can observe in our daily life are the following:

  • When the cables are connected to an amplifier, not all the existing holes will be connected to one of the cables, but some will, and there will never be two connections in the same hole.
  • When determining user identifiers, each of these users will have a unique identifier and no two users can have the same id.

They are also injective functions:

  • The area of ​​a square: f (side) = side 2
  • The length of a circumference : f (radius) = 2 π radius
  • The area of a circle : f (radius) = π radius 2
  • The cube function : f (x) = x 3

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