What is a collection? In Cantor's words, collection is to aggregate concrete or ideologically determined and different objects into a whole. Simply put, a set is a set of things. For example, "People's Republic of China (PRC) is a municipality directly under the Central Government", "People who are late for math class on Tuesday" and "Shoes worn by Zhang San" are all collections. Birds of a feather flock together. People or things of the same kind always have the same characteristics or properties. According to these characteristics or properties, a class can be determined, and this class is a set. Everyone or things are always in different collections, and even you will be a member of some collections, such as your family, your class at school, the extracurricular activity group you join, etc. In life, we sometimes use other words such as group, collection, team, race, class and stratum to express collection.
A set can be a set of numbers, a group of people, some graphics and a concept. Everything that constitutes a set belongs to this set, and the individuals belonging to this set are called elements of the set. For example, "positive odd number less than 7" is a set, and 1, 3, and 5 that make up this set are elements of this set. "Middle school textbooks" are also a collection, and the physics textbooks, chemistry textbooks and English textbooks that make up this collection are all elements of this collection. Given a set, it specifies which elements the set consists of. Obviously, for anything, either it belongs to a set or it doesn't belong to this set, the two must be one of them. For example, 1 and 3 belong to the set of "positive odd numbers less than 7", while 6 and 8 do not belong to this set.
The "positive odd number less than 7" set consists of elements 1, 3 and 5. People usually use symbols to express this statement, which is recorded as: {1, 3,5}. Braces {} indicate the element composition of the set, and generally use English capital letters to indicate the set, such as A={ 1, 3,5. This method of representing a collection is called enumeration. "Odd number less than 7" represents the set by describing the homomorphism of the elements of the set, so this method is also called description or characteristic method. These two representations are interchangeable. Generally speaking, if a set consists of poor elements, and these elements are clearly known, then the simplest method is enumeration, for example, "China's municipalities directly under the central government" can be expressed as {Beijing, Tianjin, Shanghai} ",while if there are infinite elements, or even if there are poor elements, there are too many elements, then description methods are generally used, such as" residents of Jinan "and" odd numbers greater than 9 ". Sometimes the description can also be expressed as follows: A={X|X is a resident of Jinan}, and B={X|X is an odd number greater than 9}.
In arithmetic, we often compare some numbers to find out which one is bigger. Sets can also be compared. One of the methods of comparison is to compare the elements of one set with those of another set. Sets {1, 3, 5, 7} are different from sets {2, 4, 6, 8} because their elements are different. Set A={a, b, c} and set B={c, b, a} are the same, because these two sets have the same elements, so we will record them as a = b, and it doesn't matter whether the arrangement order of the elements is the same. As long as two sets have the same elements, they are equal.
You can also use a one-to-one correspondence method to compare collections. In ancient times, a man was framed and locked in a dark basement. He is bent on going out for revenge as soon as possible, but in this dark world, there is no difference between night and day, and of course there is no concept of day. How do you know how many days you have been here? He found a trick. It turns out that the jailer emptied the toilet every other day. So whenever the jailer emptied the toilet, he drew a line on the wall with a stone, so that the set of the toilet and the set of the line formed a one-to-one correspondence, and the set of the toilet and the set of the date formed a one-to-one correspondence. So you can know the number of days from the number of rows.
To compare any two sets, just use the elements of one set to correspond to the elements of the other set. If there is a one-to-one correspondence between two sets, then we say that the two sets are equivalent, such as the set of lines, the set of toilets and the set of dates mentioned above. But it is worth noting that the equivalence and equality of two sets are not the same thing. For example, in Class Two, Grade One, there are two students, Zhang San and Li Si. Teacher Zhang San's set A is equal to teacher Li Si's set B, because the elements of the two sets are exactly the same. That is:
A={ Wu Wang, Liu Zhao, qi zhou}
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B={ Wu Wang, Liu Zhao, qi zhou}
But if Zhang San and Li Si are not in the same school, teacher Zhang San's set A and teacher Li Si's set C are not equal but equivalent, because the elements of the two sets are only one-to-one correspondence, not the same, that is:
A={ Wu Wang, Liu Zhao, qi zhou}
C={ May 8th, Zheng Jiu, Chen Shi}
The simplest way to judge whether several sets are equivalent is to see whether the number of elements in each set is equal. The number of elements in a set is called the cardinality of this set. For example, {Beijing-Tianjin-Shanghai} has three elements, so its cardinal number is 3, while {Kong Yiji, Storm, The True Story of Ah Q and One Little Thing} has four elements, so its cardinal number is 4.
Some sets have poor elements, such as {1, 4, 9, ... 100}, {Reagan, Bush, Clinton}, which are called poor sets. But some sets have infinite elements, such as sets of integers and sets of stars in the universe. This set is called an infinite set. The cardinality of an infinite set is greater than that of any finite set. From the analysis in the previous section, it can be seen that infinite sets can be compared by one-to-one correspondence, but there are surprising results, such as there are as many elements in even sets as in natural number sets, and the sets of points on straight lines are equal to those on flat surfaces. Cantor took the concept of infinite set as the basis of set theory, and proved that an obvious feature of infinite set is that it can correspond to its parts one by one.
There is also a set, which is just the opposite of an infinite set. This set does not contain any elements, such as "odd set divisible by 2" and "set of people who live to 1200 years old". These sets are called empty sets. When we discuss an object with certain properties, we call the set of all elements with the same properties as this * * * a complete set. For example, in a sports meeting, there are 10 athletes participating in a certain event, then the collection of 10 athletes is the complete works of the participating athletes.
In a collection, we can take out some elements to form a new collection. In the example mentioned at the beginning of this section, "students in Class 3, Grade 2" is a collection. Among these students, several different types of students can be divided, such as students who participate in singing competitions, students who participate in calligraphy and painting competitions, and students who participate in Go competitions. These types of students are a collection of students from Class 3, Grade 2, and they are all subsets of students from Class 3, Grade 2. Obviously, a subset is an element contained in a subset of the original set. Zhang San, for example, is not only an element in the collection of students participating in the painting and calligraphy contest, but also an element in the collection of students in Class 3, Grade 2, Senior High School. Of course, we can also form different subsets according to other conditions. For example, male students' collection, female students' collection, league members' collection, students' participation in English study groups and so on. So how many subsets can a given set make up? Let's take a concrete look, for example:
{1} can have two subsets {} and {1};
{1, 2} can have four subsets {}, {1}, {2} and {1, 2}.
{1, 2,3} can be {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2,3}, {/kloc.
By analogy, we can see that a set with n elements has 2 n subsets.
It should be noted that in set theory, there is no limit to how many elements there are in a set, so there will be a subset (empty set) with only one element or no element, and the original set itself is its own subset. So when we ask how many subsets the original set can have, the empty set and the original set must also be counted.
All subsets of a set can also form a set, which is called the power set of the original set. For example, the power set of this episode {Zhang San, Li Si} is
{{} {Zhang San}, {Li Si}, {Zhang San, Li Si}}
It is also possible to operate two or more groups to form a new group. For example, the set of excellent students in English exam is A={ Zhao Li, Wang Fang, Chen Feng}, and the set of excellent students in math exam is B={ Zhu Jun, Wang Ming, Wang Fang}. These two sets can be added to form set C, which contains both elements of A and B, and this set is {Zhao Li, Wang Fang, Chen Feng, Zhu Jun, Wang Ming}. This set is called the union of A and B. Note that in the above concentration, Wang Fang doesn't have to write it twice, just once, which means that she is an element of C. Therefore, the radix of C is not equal to the radix of A plus the radix of B, but the sum of them, and then the same element is subtracted. Yu Juan, a member of the Literary and Art Committee, actually adds up the three subsets when making statistics, but only by adding up the cardinality of the three subsets and subtracting the same elements can it be equal to the total number of Class 3, Grade 2. Meng Juan simply added it and forgot to subtract the same element. No wonder there are eight more people. Sets A and B can also be multiplied to get a new set D, which is a set composed of the same elements in A and B, namely {Wang Fang}, and D is called the intersection of A and B.
These are some basic concepts of Cantor's set theory. At that time, the German mathematics authority and his teacher Kroneck's attack was particularly fierce. He said: "Cantor has entered the hell of super poverty." He has a famous saying: "God created positive integers, and the rest are human works." In other words, people can only learn in a limited range of positive integers, and the infinite world is completely beyond people's ability. Don't even admit that Cantor is his student. In this case, Cantor has been oppressed and ostracized for a long time, and he can't get a professorship at the University of Berlin. He was depressed, had a nervous breakdown, gave up his math study and finally died in a mental hospital.
However, the establishment of Cantor's set theory is a milestone in the development history of human thinking, which indicates that after thousands of years of efforts, human beings have finally basically figured out the infinite nature. Therefore, more and more people begin to realize it and successfully apply it to many other fields of mathematics. Everyone thinks that set theory is really the basis of mathematics. Moreover, due to the establishment of set theory, mathematics is "absolutely strict". At this time, the mathematics kingdom, bright spring, warm sunshine, a peaceful scene. However, just as people were jubilant and happily preparing for the "Hundred Cows Banquet", an unprecedented earthquake suddenly broke out in the land of the mathematical kingdom-a series of paradoxes were discovered in the set theory.
These paradoxes can be said to be the inevitable result of Cantor's set theory. In fact, at the end of 19, Cantor himself had discovered many contradictions in his theory, but he didn't say anything, just used it quietly.
As can be seen from the above, there are two subsets in the set of 1 elements and four subsets in the set of two elements. Generally speaking, the set of n elements has 2 n subsets, the cardinality of the set of n elements is n, and the cardinality of the set composed of all subsets is 2 n, which is obviously 2n >；; N. So there is the "Cantor Theorem": the radix of the power set of any set (including infinite set) is greater than the radix of the power set of this arbitrary set.