Set equivalence: if there is bijection between a set and a set (also called complete one-to-one mapping), we call it sum equivalence and record it as.
For example, if it's a bijection, then.
Like a double shot, so ...
"Equivalence" is a common concept in mathematics. Readers can try to verify three properties of equivalence relation:
(i) reflexivity: for any set;
(2) Symmetry: If, then;
(iii) Transitivity: if and then.
Theorem: Let a pairwise disjoint family be a pairwise disjoint family. If it exists for each one, it is proved from the conditions that the order is bijective for each one. Now, for everyone, there is only one reason, and at this time, different records are defined as bijection. Theorem proved.
Of course, except, it is sometimes used to express equivalence with, where sum is called radix or potential of sum respectively. So "two sets have the same cardinality" is another way of saying that two sets are equivalent.
If there is a positive integer that makes a set equal to, it is called a finite set; Otherwise it is called infinite set. Especially if (all positive integers) is called countable set. Obviously, to be countable, the necessary and sufficient condition is the shape in which all elements in A can be arranged.
Finite sets and countable sets are collectively called at most countable sets.
Theorem: (i) Any infinite set must contain countable sets;
(ii) Any infinite subset of a countable set is countable;
(iii) The union of at most several sets is countable.
It is proved that (i) is set as an infinite set. If it is taken, it is an infinite set. Take is an infinite set. Take again, and so on. So we got it.
(ii) Let be an infinite subset of countable sets. Because it is an infinite set, infinite positive integers are obtained by the above method. It is easy to know that this is a countable set.
(iii) We only prove countability if countable disjoint countable sets are set. At this time, for each, now, for each, the order is obviously finite set and countable set. However, it is a countable set. Theorem proved.
Note: The proof method of (iii) is called diagonal rule. Although it is not easy to understand intuitively, it is given a more intuitive understanding in the book Basic Principles of Mathematical Analysis. The picture below is the picture in the book: in short, the row is the first countable set arrangement, which can be arranged in a row in the order of the arrows in the picture. Of course, it is not difficult to find that the number transmitted by the first arrow from the upper left to the lower right is actually in the process of proof.
Example: The elements in any disjoint open interval family can be counted at most.
In fact, rational numbers are desirable for everyone.
Because the elements in are disjoint, it is equivalent to a subset of rational numbers. We prove that rational numbers are countable, so the subset of rational numbers is countable at most. Therefore, the element in A can only be counted as one at most.