In mathematics, there is the concept of "set", as well as there are examples of comparing these same sets with each other. The names of types of comparison of sets are the following words: bijection, injection, surjection. Each of them is described in more detail below.
A bijection is… what is it?
One group of elements of the first set is matched with the second group of elements from the second set in this form: each one element of the first group is directly matched with another one element of the second group, and there is no situation with a shortage or enumeration of elements of any or from two groups of sets.
Formulation of the main properties:
- One element to one.
- There are no extra elements when matching and the first property is preserved.
- It is possible to reverse the mapping while maintaining the general view.
- A bijection is a function that is both injective and surjective.
Bijection from the scientific point of view
Bijective functions are exactly isomorphisms in the category "set and set of functions". However, bijections are not always isomorphisms for more complex categories. For example, in a certain category of groups, morphisms must be homomorphisms, since they must preserve the structure of the group. Therefore, isomorphisms are group isomorphisms, which are bijective homomorphisms.
The concept of "one-to-one correspondence" is generalized to partial functions, where they are called partial bijections, although a partial bijection is what should be an injection. The reason for this relaxation is that the partial (proper) function is no longer defined for part of its domain. Thus, there is no good reason to limit its inverse function to a complete one, i.e., defined everywhere in its domain. The set of all partial bijections to a given base set is called a symmetric inverse semigroup.
Another way of defining the same concept: it is worth saying that a partial bijection of sets from A to B is any relation R (partial function) with the property that R is a bijection graph f:A'→B ' where A' is a subset of A and B' is a subset of B.
When a partial bijection is on the same set, it is sometimes called a one-to-one partial transformation. An example is the Möbius transform just defined on the complex plane, not its completion in the extended complex plane.
Injection
One group of elements of the first set is matched with the second group of elements from the second set in this form: each one element of the first group is matched with another one element of the second, but not all of them are converted into pairs. The number of unpaired elements depends on the difference in the number of these very elements in each of the sets: if one set consists of thirty-one elements, and the other has seven more, then the number of unpaired elements is seven. Directed injection into the set. Bijection and injection are similar, but nothing more than similar.
Surjection
One group of elements of the first set is matched with the second group of elements from the second set in this way: each element of any group forms a pair, even if there is a difference between the number of elements. It follows that one element from one group can pair with several elements from another group.
Neither bijective, nor injective, nor surjective function
This is a function of bijective and surjective form, but with a remainder (unpaired)=> injection. In such a function, there is clearly a connection between bijection and surjection, since it directly includes these two types of set comparisons. So, the totality of all kinds of these functions is not one of them in isolation.
Explanation of all kinds of functions
For example, the observer is fascinated by the following. There are archery competitions. Each ofparticipants wants to hit the target (in order to facilitate the task: exactly where the arrow hits is not taken into account). Only three participants and three targets - this is the first site (site) for the tournament. In subsequent sections, the number of archers is preserved, but the number of targets is changed: on the second - four targets, on the next - also four, and on the fourth - five. Each participant shoots at each target.
- The first venue for the tournament. The first archer hits only one target. The second hits only one target. The third one repeats after the others, and all the archers hit different targets: those that are opposite them. As a result, 1 (the first archer) hit the target (a), 2 - in (b), 3 - in (c). The following dependence is observed: 1 – (a), 2 – (b), 3 – (c). The conclusion will be the judgment that such a comparison of sets is a bijection.
- The second platform for the tournament. The first archer hits only one target. The second also hits only one target. The third one doesn't really try and repeats everything after the others, but the condition is the same - all the archers hit different targets. But, as mentioned earlier, there are already four targets on the second platform. Dependence: 1 - (a), 2 - (b), 3 - (c), (d) - unpaired element of the set. In this case, the conclusion will be the judgment that such a set comparison is an injection.
- The third venue for the tournament. The first archer hits only one target. The second one hits only one target again. The third decides to pull himself together and hits the third and fourth targets. As a result, the dependence: 1 -(a), 2 - (b), 3 - (c), 3 - (d). Here, the conclusion will be the judgment that such a comparison of sets is a surjection.
- The fourth platform for the tournament. With the first, everything is already clear, he hits only one target, in which there will soon be no room for already boring hits. Now the second takes on the role of the still recent third and again hits only one target, repeating after the first. The third continues to control himself and does not stop introducing his arrow to the third and fourth targets. The fifth, however, was still beyond his control. So, dependence: 1 - (a), 2 - (b), 3 - (c), 3 - (d), (e) - unpaired element of the set of targets. Conclusion: such a comparison of sets is not a surjection, not an injection, and not a bijection.
Now constructing a bijection, injection or surjection will not be a problem, as well as finding differences between them.
