In mathematics, there is the concept of "set", just as there are examples of comparing these same sets with each other. The names of the types of matching sets are the following words: bijection, injection, surjection. Below each of them is described in more detail.
A bijection is ... is that what?
One group of elements of the first set is mapped to the second group of elements from the second set in this form: each one element of the first group is directly mapped to another one element from the second group, and there is no situation with a shortage or enumeration of elements of any of the two groups of sets .
The wording of the main properties:
- One item to one.
- There are no unnecessary elements when matching and the first property is saved.
- It is possible to reverse the mapping with maintaining the general appearance.
- Bijection is such a function that it is both injective and surjective.
Scientific bijection
Bijective functions are precisely 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, which are bijective homomorphisms.
The concept of “one-to-one correspondence” is generalized to partial functions, where they are called partial bijections, although partial bijection is what should be an injection. The reason for this relaxation is that the partial (correct) function is no longer defined for part of its area. Thus, there is no good reason to limit its inverse function to a complete one, i.e., defined everywhere in its field. The set of all partial bijections onto a given basic 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 Moebius transform, simply defined on the complex plane, and not its completion in the extended complex plane.
Injection
One group of elements of the first set is compared with the second group of elements from the second set in this form: each one element of the first group is compared 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 same elements in each of the sets: if one set consists of thirty-one elements, and in the other seven more, then the number of unpaired elements is seven. Directed injection into the set. Bijection and injection are similar to each other, but no more than just similar.
Surjection
One group of elements of the first set is compared with the second group of elements from the second set in this form: each element of a group forms a pair, even if there is a difference between the number of elements. It follows that one element from one group can create a pair with several elements from another group.
Neither bijective, nor injective, nor surjective function
This is a bijective and surjective function, but with a residual element (unpaired) => injection. In such a function, there is clearly a connection between bijection and surjection, since it directly includes these two types of matching sets. So, the totality of all types of these functions is not one of them individually.
Explanation of all kinds of functions
For example, the observer is keen on the following. Archery competitions are taking place. Each of the participants wants to hit the target (in order to facilitate the task: where exactly the arrow hits is not taken into account). Only three participants and three targets - this is the first platform (site) for the tournament. In subsequent sections, the number of archers remains, but the number of targets changes: in the second - four targets, in the next - also four, and in the fourth - five. Each participant shoots at each target.
- The first platform for the tournament. The first archer hits only one target. The second hits only one target. The third 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 juxtaposition 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 does not particularly try and repeats everything after others, but the condition is the same - all archers hit different targets. But, as mentioned earlier, there are already four targets on the second site. 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 mapping of sets is an injection.
- The third platform for the tournament. The first archer hits only one target. The second again hits only one target. 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 surjection.
- The fourth platform for the tournament. With the first, everything is already clear, it hits only one target, in which there will soon be no room for already bored hits. Now the second takes on the role of the more recent third and again hits only one target, repeating after the first. The third continues to control himself and does not cease to acquaint his arrow with the third and fourth target. The fifth, however, nevertheless proved to be beyond his control. So, the dependence: 1 - (a), 2 - (b), 3 - (c), 3 - (d), (e) is an unpaired element of a set of targets. Conclusion: such a comparison of sets is not surjection, not an injection, and not a bijection.
Now to build a bijection, injection or surjection will not be a problem, as well as to find differences between them.