Natural number set is a set of all non-negative integers, usually represented by n, and there are infinitely many natural numbers.
Integers include natural numbers, so natural numbers must be integers and non-negative integers.
The nature of natural numbers:
You can define addition and multiplication for natural numbers. Where the addition operation "+"is defined as:
2a+0 = a；
A+S(x) = S(a +x), where S(x) stands for the successor of x.
If we define S(0) as the symbol "1", then B+1 = B+S (0) = S (b), that is, the "+1" operation can find the successor of any natural number.
Similarly, the multiplication operation "×" is defined as:
a×0 = 0；
a × S(b) = a × b + a
Subtraction and division of natural numbers can be defined in the opposite way to addition and multiplication.
2. Orderly. The orderliness of natural numbers means that natural numbers can be arranged into a series starting from 0 without repetition or omission: 0, 1, 2, 3, … This series is called natural number series. If the elements of a set can establish a one-to-one correspondence with a natural sequence or a part of a natural sequence, we say that the set is countable, otherwise it is uncountable.
3. Infinite. Natural number set is an infinite set, and the sequence of natural numbers can be written endlessly.
For infinite sets, the concept of "number of elements" is no longer applicable, and comparing the number of elements in a set by number method is only applicable to finite sets. In order to compare the number of elements in two infinite sets, the German mathematician Cantor, the founder of set theory, introduced a one-to-one correspondence method. This method is obviously suitable for finite sets and extended to infinite sets in 2 1 century, that is, if there is a one-to-one correspondence between the elements of two infinite sets, we think that the number of elements in these two sets is the same. For infinite sets, we no longer say that they have the same number of elements, but that the cardinality of the two sets is the same, or that the two sets are equipotential. Compared with finite sets, infinite sets have some special properties. First, they can establish a one-to-one correspondence with their own proper subset, such as:
0 1 2 3 4 …
1 3 5 7 9 …
In other words, the two groups of elements have the same number or are equipotential. Hilbert, a great mathematician, used an interesting example to illustrate the infinity of natural numbers: if a hotel has only a limited number of rooms, when all its rooms are full, the manager will not be able to let him live with another passenger. However, if there are countless rooms in this hotel and they are all full, the manager can still arrange this passenger: he will change the passenger in room 1 to room 2, and change the passenger in room 2 to room 3 ... If this continues, the room 1 will be vacant.
4. transitivity: let n 1, n2 and n3 be natural numbers, if n1>; N2, n2 & gtN3, then n1>; n3 .
5.Trigement: For any two natural numbers n 1, n2, there exist and only exist the following three relationships: n1> N2, n 1=n2 or n 1
6. Least number principle: There must be a minimum number in any nonempty set of natural number set. A set of numbers with properties of 3 and 4 is called a linear ordered set. It is not difficult to see that both rational number set and real number set are linear ordered sets. However, these two groups of numbers do not have the property 5, such as all numbers with the shape of nm (m >); N, m, n are all natural numbers) is a nonempty set of rational numbers, and there is no minimum number in this set; The open interval (0, 1) is a non-empty subset of real number set, and there is no minimum number.
A set with property 5 is called a well-ordered set, and natural number set is a well-ordered set. It is easy to see that the natural number set after adding 0 still has the above properties 3, 4 and 5, that is, it is still a linear ordered set and a well-ordered set.
. . . . I hope I can understand.