Injection function definition - What is it, Meaning and Concept

In the context of mathematics , the link that develops between two sets is called function , through which each element of a set is assigned a only element of another set or none.The idea of ​​ injective or injective , on the other hand, refers to the property that indicates that two different elements of a first set correspond to them two other different elements of a second set.

An injective function , therefore, is that which, to different elements of the initial set (the domain ), different elements of the final set correspond to them (the codomain).This means that each element of the codomain has no more than one preimage in the domain: or, expressed in another way, that each element of the domain cannot have more than one image in the co-domain .

The expression of an injective function is f: x-> y .Take the case of a X set consisting of Argentina , Switzerland and Nigeria , and a Y set consisting of America , Europe and Africa .If we would like to establish a relationship Between each country and its corresponding continent, we would obtain an injective function, since the links would be the following:

Argentina-> AmericaSwitzerland-> EuropeNigeria-> Africa

With the mentioned sets and the relationship listed, the elements of the first set (the countries ) could never correspond to more than one image in the second set (the continents). Argentina belongs to America , and not to Europe or Africa . Switzerland , meanwhile, this only in Europe (not in America or in Africa ). Nigeria , finally, it is only part of Africa , without being in America or in Europe .In this case, in short, both sets are linked by an injection function.

Let's see below an example in which the requirements are not met so that the function can be considered injective.This is the case of the function that supports all real numbers and is defined as f (x)=xx : since it is possible to use both negative and positive numbers to replace the x variable, each result (which by convention is represented by the variable and ) can be obtained with any number and its opposite, such as 8 and -8 (for both, the result is 64 ).

This is not possible with examples such as the one that involves countries and their continents, but this does not mean that there are no relationships less than mathematics strict or, so to speak, more flexible.If we think of a set in which the names of ten people and another are listened to, their co-domain, in which some of their friends are, it would be possible that for each element of the second there were more than one of the domain.

Returning to the scope of the numbers, if we wanted to alter the previous function so that it would become injective we should only restrict the domain to the positive real numbers: in this way, never a element of one of the sets would be related to more than one of the other.

The formal definition of injective function is as follows: f: X-> Y is injective only if for the elements of the set X a and b it is true that f (a) is equal to f (b) when a is equal to b .In other words, the function is also injective if when the elements are different, so are your images .

On the other hand, if we have two sets among which there is an injective function we speak of cardinality when for the elements of the first they are less than or equal to their images.If a second function will relate the sets in the opposite direction, then one would say that there is a bijective application between the sets.