In the context of math, is named function the link that develops between two sets, through which each element of a set is assigned a single element of another set or none at all. The idea of injective or injective, on the other hand, it alludes to the property that indicates that two different elements of a first set correspond to two different elements of a second set.
A injective function, therefore, is one that, to different elements of the initial set (the domain), different elements of the final set (the codomain) correspond to them. This means that each element of the codomain has no more than one preimage in the domain: or, expressed in another way, that each domain element cannot have more than one image in the codomain.
The expression of an injective function is f: x -> y. Take the case of a set X formed by Argentina, Swiss Y Nigeria, and a set Y composed by America, Europe Y Africa. If we wanted 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 -> America
Switzerland -> Europe
Nigeria -> Africa
With the mentioned sets and the indicated relationship, the elements of the first set (the countries) could never correspond to more than one image in the second set (the continents). Argentina belonging to America, and not to Europe nor to Africa. Swiss, meanwhile, is alone in Europe (not in America nor 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 injective function.
Let’s see below an example in which the requirements for the function to be considered injective are not met. Such 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 variable x, each result (which by convention is represented by the variable Y) can be obtained with any number and its opposite, such as 8 Y -8 (for both, the result is 64).
This is not possible with examples such as the one involving countries and their continents, but this does not mean that outside of mathematics there is no relations less strict or, as it were, more flexible. If we think of a set in which the names of ten people are listed and another, their codomain, in which some of their friends are, it would be possible that for each element of the second there would be more than one of the domain.
Returning to the scope of numbers, if we wanted to alter the previous function so that it became injective, we would only have to restrict the domain to positive real numbers: in this way, no longer a element of one of the sets would be related to more than one of the other.
The formal definition of an injective function is as follows: f: X -> Y it is injective only if for the elements of the set X to Y b It is true that fa) is equal to f (b) when to is equal to b. In other words, the function is also injective if when the elements are different, so are their images.
On the other hand, if we have two sets between which there is an injective function, we speak of cardinality when the elements of the first are less than or equal to their images. If a second function related the sets in the reverse sense, then it would be said that there are a bijective application between sets.