Sets Elements of the theory of sets. Operations on sets
I don’t remember when I first learned about topology, but this science immediately interested me. The teapot turns into a donut, the sphere turns inside out. Many have heard about it. But those who want to delve into this topic on a more serious level often have difficulties. This is especially true for mastering the most basic concepts, which are inherently very abstract. Moreover, many sources seem to deliberately seek to confuse the reader. Let's say the Russian wiki gives a very vague formulation of what topology does. It says that it is a science that studies topological spaces. In the article on topological spaces, the reader can learn that topological spaces are spaces equipped with topology. Such explanations in the style of Lemov's sepules do not really clarify the essence of the subject. I will try further to state the main basic concepts in a clearer form. In my note, there will be no transforming teapots and bagels, but the first steps will be taken that will eventually allow you to learn this magic.
However, since I am not a mathematician, but a 100% humanist, it is quite possible that what is written below is a lie! Well, or at least part of it.
I first wrote this note, as the beginning of a series of articles on topology, for my humanitarian friends, but none of them began to read it. I decided to post the corrected and extended version on Habr. It seemed to me that there is a certain interest in this topic here and there have not yet been articles of this kind. Thanks in advance for all comments about errors and inaccuracies. Be warned that I use a lot of pictures.
Let's start with a brief recap of set theory. I think most readers are familiar with it, but nevertheless I will remind you of the basics.
So, it is believed that the set has no definition and that we intuitively understand what it is. Kantor said this: “Under the “set” we mean the combination into a certain whole M of certain well-distinguished objects m of our contemplation or our thinking (which will be called the “elements” of the set M)”. Of course, this is just an allegorical description, not a mathematical definition.
Set theory is known for (pardon the pun) a lot of amazing paradoxes. For example . It is also associated with the crisis of mathematics at the beginning of the 20th century.
Set theory exists in several variants such as ZFC or NBG and others. A variant of the theory is type theory, which is quite important to programmers. Finally, some mathematicians suggest using category theory as the foundation of mathematics instead of set theory, about which much has been written on Habré. Type theory and set theory describe mathematical objects as if "from the inside", while category theory is not interested in their internal structure, but only in how they interact, i.e. gives their "external" characteristics.
For us, only the very basic foundations of set theory are important.
Sets are finite.
They are endless. For example, a set of integers, which is denoted by the letter ℤ (or just Z, if you don't have curly letters on your keyboard).
Finally, there is an empty set. It is exactly one in the entire universe. There is a simple proof of this fact, but I will not give it here.
If the set is infinite, it happens countable. Countable - those sets whose elements can be renumbered by natural numbers. The set of natural numbers itself, you guessed it, is also countable. Here's how to enumerate integers.
With rational numbers it is more difficult, but they can also be numbered. This method is called diagonal process and looks like the picture below.
We zigzag through rational numbers, starting from 1. At the same time, we assign an even number to each number that we get. Negative rational numbers are counted in the same way, only the numbers are odd, starting with 3. Zero traditionally gets the first number. Thus, it is clear that all rational numbers can be numbered. All numbers like 4.87592692976340586068 or 1.00000000000001 or -9092 or even 42 get their number in this table. However, not all numbers are included here. For example, √2 will not get a number. At one time, this greatly upset the Greeks. They say the guy who discovered irrational numbers got drowned.
A generalization of the concept of size for sets is power. The cardinality of finite sets is equal to the number of their elements. The cardinality of infinite sets is denoted by the Hebrew letter aleph with an index. The smallest infinite power is the power ℵ 0 . It is equal to the cardinality of countable sets. As we see, therefore, there are as many natural numbers as there are integers or rational ones. Strange, but true. Next is power continuum. It is denoted by a small Gothic letter c. This is the cardinality of the set of real numbers ℝ, for example. There is a hypothesis that the power of the continuum is equal to the power ℵ 1 . That is, that this is the next cardinality after the cardinality of countable sets, and there is no intermediate cardinality between countable sets and the continuum.
You can perform various operations on sets and get new sets.
1. Sets can be combined.
3. You can search for the intersection of sets.
Actually, this is all about sets that you need to know for the purposes of this note. Now we can proceed to the topology itself.
Topology is the science that studies sets with a specific structure. This structure is also called a topology.
Let us have some non-empty set S.
Let this set have some structure, which is described using a set, which we will call T. T is a set of subsets of the set S such that:
1. S itself and ∅ belong to T.
2. Any union of arbitrary families of elements T belongs to T.
3. Intersection of an arbitrary final of the element family T belongs to T.
If these three points hold, then our structure is the topology of T on the set S. The elements of the set T are called open sets on S in the topology T. The complement to open sets are closed sets. It is important to note that if a set is open, this does not mean that it is not closed and vice versa. In addition, in a given set, with respect to some topology, there may be subsets that are neither open nor closed.
Let's take an example. Suppose we have a set consisting of three colored triangles.
The simplest topology on it is called antidiscrete topology. Here she is.
This topology is also called the topology sticky dots. It consists of the set itself and the empty set. This indeed satisfies the axioms of topology.
Multiple topologies can be defined on one set. Here is another very primitive topology that happens. It's called discrete. It is a topology that consists of all subsets of a given set.
And here is the topology. It is given on a set of 7 multi-colored stars S, which I have marked with letters. Make sure it's a topology. I'm not sure about this, suddenly I missed some kind of union or intersection. This picture should contain the set S itself, the empty set, the intersections and unions of all other elements of the topology should also be in the picture.
Pair from the topology and the set on which it is given is called topological space.
If there are many points in the set (not to mention the fact that there can be infinitely many of them), then listing all open sets can be problematic. For example, for a discrete topology on a set of three elements, you need to make a list of 8 sets. And for a 4-element set, the discrete topology will already have 16, for 5 - 32, for 6 -64, and so on. In order not to enumerate all open sets, a kind of abbreviated notation is used - those elements are written out, the unions of which can give all open sets. It is called base topology. For example, for a discrete space topology of three triangles, these will be three triangles taken separately, because by combining them, you can get all the other open sets in this topology. The base is said to generate the topology. A set whose elements generate a base is called a prebase.
Below is an example of a base for a discrete topology on a set of five stars. As you can see, in this case the base consists of only five elements, while the topology has as many as 32 subsets. Agree, using the base to describe the topology is much more convenient.
What are open sets for? In a sense, they give an idea of the "proximity" between points and the difference between them. If the points belong to two different open sets, or if one point is in an open set that does not contain the other, then they are topologically different. In the antidiscrete topology, all points are indistinguishable in this sense, they seem to stick together. Conversely, in the discrete topology All points are different.
The notion of an open set is inextricably linked with the notion neighborhood. Some authors define topology not in terms of open sets, but in terms of neighborhoods. The neighborhood of the point p is the set that contains the open ball centered at this point. For example, the figure below shows neighborhoods and non-neighborhoods of points. The set S 1 is a neighborhood of the point p, but the set S 2 is not.
The relationship between open set and octestity can be formulated as follows. An open set is such a set, each element of which has some neighborhood lying in the given set. Or vice versa, one can say that a set is open if it is a neighborhood of any of its points.
All these are the most basic concepts of topology. From here it is not yet clear how to turn the spheres inside out. Maybe in the future, I will be able to get to this kind of topics (if I figure it out myself).
UPD. Due to the inaccuracy of my speech, there was some confusion about the cardinalities of the sets. I have slightly corrected my text and here I want to give an explanation. Kantor, creating his theory of sets, introduced the concept of cardinality, which made it possible to compare infinite sets. Cantor established that the cardinalities of countable sets (for example, rational numbers) and continuum (for example, real numbers) are different. He suggested that the cardinality of the continuum is next after the cardinality of countable sets, i.e. equals aleph-one. Cantor tried to prove this conjecture, but without success. Later it became clear that this hypothesis could neither be disproved nor proved.
The concept of a set is the original not strictly defined concept. Let us give here the definition of a set (more precisely, an explanation of the idea of a set) belonging to G. Cantor: “Under a variety or a set, I mean in general all the many things that can be thought of as a single one, i.e. such a set of certain elements that can be connected by means of one law into one whole."
Sets will, as a rule, be denoted by capital letters of the Latin alphabet, and their elements by small letters, although sometimes this convention will have to be deviated from, since the elements of a certain set may be other sets. The fact that the element a belongs to the set is written as .
In mathematics, we deal with a wide variety of sets. For the elements of these sets, we use two main types of notation: constants and variables.
An individual constant (or just a constant) with a range denotes a fixed element of a set. Such, for example, are the designations (records in a certain number system) of real numbers:. For two constants and with a range of values, we will write , meaning by this the coincidence of the elements of the set denoted by them.
An individual variable (or just a variable) with a range denotes an arbitrary, not predetermined element of the set. In this case, they say that the variable runs through the set or the variable takes arbitrary values on the set. You can fix the value of a variable by writing , where is a constant with the same range as . In this case, they say that instead of a variable, its specific value was substituted, or a substitution was made instead, or the variable took on the value .
Equality of variables is understood as follows: whenever a variable takes on an arbitrary value, the variable takes on the same value, and vice versa. Thus, equal variables "synchronously" always take on the same values.
Usually, constants and variables whose range is a certain numerical set, namely one of the sets and , are called natural, integer (or integer), rational, real, and complex constants and variables, respectively. In the course of discrete mathematics, we will use various constants and variables, the range of which is not always a numerical set.
To shorten the record, we will use logical symbolism, which allows us to write statements briefly, like formulas. The concept of an utterance is not defined. It is only indicated that any statement can be true or false (of course, not both at the same time!).
Logical operations (bindings) on sets
To form new statements from existing statements, the following logical operations (or logical connectives) are used.
1. Disjunction: a statement (read: "or") is true if and only if at least one of the statements and is true.
2. Conjunction: a statement (read: "and") is true if and only if both statements and are true.
3. Negation: a statement (read: "not") is true if and only if it is false.
4. Implication: a statement (read: "if, then" or "implies") is true if and only if the statement is true or both statements are false.
5. Equivalence (or equivalence): a statement (read: "if and only if") is true if and only if both statements and are either simultaneously true or simultaneously false. Any two statements and such that is true are called logically equivalent or equivalent.
Writing statements using logical operations, we assume that the order in which all operations are performed is determined by the arrangement of brackets. To simplify notation, parentheses are often omitted, while accepting a certain order of operations ("priority convention").
The negation operation is always performed first, and therefore it is not enclosed in parentheses. The second one performs the operation of conjunction, then disjunction, and finally implication and equivalence. For example, a statement is written like this: This statement is a disjunction of two statements: the first is a negation, and the second is. In contrast, the proposition is the negation of the disjunction of the propositions and .
For example, the statement after placing brackets in accordance with priorities will take the form
Let us make some comments about the logical connectives introduced above. The meaningful interpretation of disjunction, conjunction and negation does not need special explanations. An implication is true, by definition, whenever the proposition is true (regardless of truth) and both are both false. Thus, if the implication is true, then when true, the truth takes place, but the opposite may not be true, i.e. if false, the statement can be either true or false. This motivates the reading of the implication in the form of "if , then". It is also easy to understand that the proposition is equivalent to the proposition and thus meaningfully "if , then" is identified with "not or".
Equivalence is nothing but a "two-sided implication", i.e. is tantamount to . This means that truth follows from truth and, conversely, truth follows from truth.
Example 1.1. To determine the truth or falsity of a complex statement, depending on the truth or falsity of the statements included in it, truth tables are used.
The first two columns of the table contain all possible sets of values that the statements and can take. The truth of the statement is indicated by the letter "I" or the number 1, and the falsity - by the letter "L" or the number 0. The remaining columns are filled in from left to right. So for each set of values and find the corresponding values of statements.
The truth tables of logical operations have the simplest form (Tables 1.1-1.5).
Consider a complex statement. For the convenience of calculations, we denote the statement by , the statement by , and write the original statement as . The truth table of this statement consists of columns and (Table 1.6).
Predicates and quantifiers
Compound statements are formed not only through logical connectives, but also with the help of predicates and quantifiers.
A predicate is a statement containing one or more individual variables. For example, "there is an even number" or "there is a student of Moscow State Technical University named after Bauman, who entered in 1999". In the first predicate there is an integer variable, in the second - a variable running through the set of "human individuals". An example of a predicate containing several individual variables is: "has a son", "and study in the same group", "is divided by", "is less than", etc. Predicates will be written in the form , assuming that all variables included in the given predicate are listed in brackets.
Substituting a specific value for each variable included in the predicate , i.e. fixing the values , where are some constants with the corresponding range of values, we obtain a statement that does not contain variables. For example, "2 is an even number", "Isaac Newton is a student of the Moscow State Technical University named after Bauman, who entered in 1999", "Ivanov is the son of Petrov", "5 is divisible by 7", etc. Depending on whether the statement thus obtained is true or false, the predicate is said to be satisfied or not satisfied on the set of values of the variables . A predicate that is satisfied on any set of variables included in it is called identically true, and a predicate that is not satisfied on any set of values of its variables is called identically false.
A statement from a predicate can be obtained not only by substituting the values of its variables, but also by means of quantifiers. Two quantifiers are introduced - existence and universality, denoted by and respectively.
The proposition ("for every element in the set is true", or, more briefly, "for all is true") is true, by definition, if and only if the predicate is true for every value of the variable .
The statement ("there is, or there is, such an element of the set that is true", also "for some is true") is true, by definition, if and only if the predicate is satisfied on some values of the variable.
Associating predicate variables with quantifiers
When a statement is formed from a predicate by means of a quantifier, it is said that the variable of the predicate is bound by the quantifier. Similarly, variables are bound in predicates containing several variables. In the general case, expressions of the form
where any of the quantifiers or can be substituted for each letter with an index.
For example, the statement reads like this: "for everyone there is such that is true". If the sets that run through the variables of the predicates are fixed (meaning "by default"), then the quantifiers are written in an abbreviated form: or .
Note that many mathematical theorems can be written in a form similar to the statements with quantifiers just given, for example: "it is true for everyone and for everyone: if is a function that is differentiable at a point, then the function is continuous at a point".
Ways of specifying sets
Having discussed the features of the use of logical symbolism, let us return to the consideration of sets.
Two sets and are considered equal if any element of the set is an element of the set and vice versa. It follows from the above definition of equal sets that a set is completely determined by its elements.
Let us consider ways of specifying concrete sets. For a finite set, the number of elements of which is relatively small, the method of direct enumeration of elements can be used. The elements of a finite set are listed in curly braces in an arbitrary fixed order. We emphasize that since the set is completely determined by its elements, then when specifying a finite set, the order in which its elements are listed does not matter. Therefore, entries, etc. all define the same set. In addition, sometimes repetitions of elements are used in the notation of sets. We will assume that the entry defines the same set as the entry .
In the general case, for a finite set, the notation is used. As a rule, repetition of elements is avoided. Then the finite set given by the notation consists of elements. It is also called an n-element set.
However, the method of specifying a set by directly enumerating its elements is applicable in a very narrow range of finite sets. The most general way to specify concrete sets is to specify some property that all elements of the described set must have, and only they.
This idea is implemented in the following way. Let the variable range over some set called the universal set. We assume that only such sets are considered whose elements are also elements of the set . In this case, a property that only the elements of a given set have can be expressed by means of a predicate , which is executed if and only if the variable takes an arbitrary value from the set . In other words, true if and only if the individual constant is substituted for .
The predicate is called in this case the characteristic predicate of the set, and the property expressed with the help of this predicate is called the characteristic property or collectivizing property.
The set defined through the characteristic predicate is written in the following form:
For example, it means that "there is a set consisting of all elements such that each of them is an even natural number".
The term "collectivizing property" is motivated by the fact that this property allows you to collect disparate elements into a single whole. Thus, the property that defines a set (see below) literally forms a kind of "collective":
If we return to Cantor's definition of a set, then the characteristic predicate of a set is the law by which a set of elements is combined into a single whole. A predicate specifying a collectivizing property can be identically false. A set defined in this way will have no elements. It is called the empty set and denoted by .
In contrast, an identically true characteristic predicate defines a universal set.
Note that not every predicate expresses some collectivizing property.
Remark 1.1. The specific content of the concept of a universal set is determined by the specific context in which we apply set-theoretic ideas. For example, if we deal only with various numerical sets, then the set of all real numbers can appear as a universal one. Each branch of mathematics deals with a relatively limited set of sets. Therefore, it is convenient to assume that the elements of each of these sets are also the elements of some universal set "embracing" them. By fixing the universal set, we thereby fix the range of values of all the variables and constants that appear in our mathematical reasoning. In this case, it is precisely possible not to indicate in the quantifiers the set that runs through the variable bound by the quantifier. In what follows, we will meet with various examples of concrete universal sets.
Definition 1.many is a collection of some objects united into one whole according to some ‒ or attribute.
The objects that make up a set are called its elements.
They are denoted by capital letters of the Latin alphabet: A, B, …, X, Y, …, and their elements are denoted by the corresponding capital letters: a, b, …, x, y.
Definition 1.1. A set that does not contain any element is called empty and is denoted by the symbol Ø.
A set can be specified by enumeration and description.
Example:; 
.
Definition 1.2. many A called a subset B if each element of the set A is an element of the set B. Symbolically, this is expressed as follows: AB (A contained in B).
Definition 1.3. Two sets A And B called equal, if they consist of the same elements :( A =B).
Operations on sets.
Definition 1.4. Union or sum of sets A And B is a set consisting of elements, each of which belongs to at least one of these sets.
The union of sets is denoted AB(or A +B). Briefly, one can write AB = .
AB= A +B
If BA, That A +B=A
Definition 1.5. Intersection or product of sets A And B is called a set consisting of elements, each of which belongs to the set A and many B simultaneously. The intersection of sets is denoted AB(or A· B). Briefly, you can write:
AB= 
.
AB =A · B
If B A, That A · B=B
Definition 1.6. set difference A And B a set is called, each element of which is an element of the set A
and is not an element of the set B. The difference of sets is denoted A\B. A-priory A\B
=
.
A\B = A–B
Sets whose elements are numbers are called numerical.
Examples of numeric sets are:
N
=
is the set of natural numbers.
Z= - set of integers.
Q=
is the set of rational numbers.
R is the set of real numbers.
A bunch of R contains rational and irrational numbers. Any rational number is expressed either as a finite decimal fraction or as an infinite periodic fraction. Thus, ;… are rational numbers.
An irrational number is expressed as an infinite non-periodic decimal fraction. So, = 1.41421356...; = 3.14159265.... is an irrational number.
K is the set of complex numbers (of the form Z=a+ bi)
RK
Definition 1.7.Ɛ ‒ neighborhood of a point x 0 is called a symmetric interval ( x 0 – Ɛ; x 0 + Ɛ) containing a dot x 0 .
In particular, if the interval ( x 0 –Ɛ; x 0 +Ɛ), then the inequality x 0 –Ɛ<x<x 0 +Ɛ, or equivalently, │ x– x 0 │<Ɛ. Executing the latter means hitting the dot x in Ɛ – neighborhood of the point x 0 .
Example 1:
(2 - 0.1; 2 + 0.1) or (1.9; 2.1) - Ɛ - neighborhood.
│x– 2│< 0,1
–0,1<x – 2<0,1
2 –0,1<x< 2 + 0,1
1,9<x< 2,1
Example 2:
A– set of divisors 24;
B is the set of divisors 18.
I am a theoretical physicist by education, but I have a good mathematical background. In the magistracy one of the subjects was philosophy, it was necessary to choose a topic and submit a paper on it. Since most of the options were more than once obmusoleny, I decided to choose something more exotic. I do not pretend to novelty, I just managed to accumulate all / almost all available literature on this topic. Philosophers and mathematicians can throw stones at me, I will only be grateful for constructive criticism.P.S. Very "dry language", but quite readable after the university program. For the most part, definitions of paradoxes were taken from Wikipedia (simplified wording and ready-made TeX markup).
Introduction
Both the set theory itself and the paradoxes inherent in it appeared not so long ago, just over a hundred years ago. However, during this period a long way has been traveled, the theory of sets, one way or another, actually became the basis of most sections of mathematics. Its paradoxes, connected with Cantor's infinity, were successfully explained literally in half a century.You should start with a definition.
What is a multitude? The question is quite simple, the answer to it is quite intuitive. A set is a set of elements represented by a single object. Cantor in his work Beiträge zur Begründung der transfiniten Mengenlehre gives a definition: by “set” we mean the combination into a certain whole M of certain well-defined objects m of our contemplation or our thinking (which will be called “elements” of the set M). As you can see, the essence has not changed, the difference is only in the part that depends on the worldview of the determinant. The history of set theory, both in logic and in mathematics, is highly controversial. In fact, Kantor laid the foundation for it in the 19th century, then Russell and the others continued the work.
Paradoxes (logic and set theory) - (Greek - unexpected) - formal logical contradictions that arise in the meaningful set theory and formal logic while maintaining the logical correctness of reasoning. Paradoxes arise when two mutually exclusive (contradictory) propositions are equally provable. Paradoxes can appear both within scientific theory and in ordinary reasoning (for example, Russell's paradox about the set of all normal sets given by Russell: "The village barber shaves all those and only those inhabitants of his village who do not shave themselves. Should he shave yourself?"). Since the formal logical contradiction destroys reasoning as a means of discovering and proving the truth (in a theory in which a paradox appears, any sentence, both true and false, is provable), the problem arises of identifying the sources of such contradictions and finding ways to eliminate them. The problem of philosophical understanding of specific solutions to paradoxes is one of the important methodological problems of formal logic and the logical foundations of mathematics.
The purpose of this work is to study the paradoxes of set theory as heirs of ancient antinomies and quite logical consequences of the transition to a new level of abstraction - infinity. The task is to consider the main paradoxes, their philosophical interpretation.
Basic paradoxes of set theory
The barber only shaves people who don't shave themselves. Does he shave himself?
Let's continue with a brief excursion into history.Some of the logical paradoxes have been known since ancient times, but due to the fact that mathematical theory was limited to arithmetic and geometry alone, it was impossible to correlate them with set theory. In the 19th century, the situation changed radically: Kantor reached a new level of abstraction in his works. He introduced the concept of infinity, thereby creating a new branch of mathematics and thereby allowing different infinities to be compared using the concept of “power of a set”. However, in doing so, he created many paradoxes. The first is the so-called Burali-Forti paradox. In the mathematical literature, there are various formulations based on different terminology and an assumed set of well-known theorems. Here is one of the formal definitions.
It can be proved that if x is an arbitrary set of ordinals, then the sum-set is an ordinal greater than or equal to each of the elements x. Suppose now that is the set of all ordinal numbers. Then is an ordinal number greater than or equal to any of the numbers in . But then and is an ordinal number, moreover, it is already strictly greater, and therefore not equal to any of the numbers in . But this contradicts the condition that is the set of all ordinal numbers.
The essence of the paradox is that when the set of all ordinal numbers is formed, a new ordinal type is formed, which was not yet among “all” transfinite ordinal numbers that existed before the formation of the set of all ordinal numbers. This paradox was discovered by Cantor himself, independently discovered and published by the Italian mathematician Burali-Forti, the latter's errors were corrected by Russell, after which the formulation acquired its final form.
Among all attempts to avoid such paradoxes and to some extent try to explain them, the idea of the already mentioned Russell deserves the most attention. He proposed to exclude from mathematics and logic impredicative sentences in which the definition of an element of a set depends on the latter, which causes paradoxes. The rule sounds like this: "no set C can contain elements m, defined only in terms of the set C, as well as elements n, assuming this set in their definition" . Such a restriction on the definition of a set allows us to avoid paradoxes, but at the same time significantly narrows the scope of its application in mathematics. In addition, this is not enough to explain their nature and reasons for their appearance, rooted in the dichotomy of thought and language, in the features of formal logic. To some extent, this limitation can be traced an analogy with what in a later period cognitive psychologists and linguists began to call "basic level categorization": the definition is reduced to the most easy-to-understand and study concept.
Assume that the set of all sets exists. In this case, it is true, that is, any set t is a subset of V. But it follows from this that the power of any set does not exceed the power of V. But by virtue of the axiom of the set of all subsets, for V, as well as for any set, there is a set of all subsets , and by Cantor's theorem, which contradicts the previous statement. Therefore, V cannot exist, which conflicts with the "naive" hypothesis that any syntactically correct logical condition defines a set, i.e. that for any formula A that does not contain y freely. A remarkable proof of the absence of such contradictions on the basis of the axiomatized Zermelo-Fraenkel set theory is given by Potter.
From a logical point of view, both of the above paradoxes are identical to the "Liar" or "The Barber": the expressed judgment is directed not only to something objective in relation to him, but also to himself. However, one should pay attention not only to the logical side, but also to the concept of infinity, which is present here. The literature refers to the work of Poincaré, in which he writes: "belief in the existence of actual infinity ... makes these non-predicative definitions necessary"" .
In general, the main points are:
- in these paradoxes, the rule is violated to clearly separate the “spheres” of the predicate and the subject; the degree of confusion is close to the substitution of one concept for another;
- usually in logic it is assumed that in the process of reasoning the subject and predicate retain their scope and content, in this case
transition from one category to another, resulting in a mismatch; - the presence of the word "all" makes sense for a finite number of elements, but in the case of an infinite number of them, it is possible to have one that
to define itself would require the definition of a set; - basic logical laws are violated:
- the law of identity is violated when the non-identity of the subject and the predicate is revealed;
- the law of contradiction - when two contradictory judgments are derived with the same right;
- the law of the excluded third - when this third has to be recognized, and not excluded, since neither the first nor the second can be recognized one without the other, because they are equally valid.
Let K be the set of all sets that do not contain themselves as their element. Does K contain itself as an element? If yes, then, by definition of K, it should not be an element of K - a contradiction. If not - then, by definition of K, it must be an element of K - again a contradiction. This statement is logically derived from Cantor's paradox, which shows their relationship. However, the philosophical essence manifests itself more clearly, since the “self-movement” of concepts takes place right “before our eyes”.
Tristram Shandy's paradox:
In Stern's The Life and Opinions of Tristram Shandy, Gentleman, the hero finds that it took him a whole year to recount the events of the first day of his life, and another year to describe the second day. In this regard, the hero complains that the material of his biography will accumulate faster than he can process it, and he will never be able to complete it. “Now I maintain,” Russell objects, “that if he lived forever and his work would not become a burden to him, even if his life continued to be as eventful as at the beginning, then not one part of his biography would not remain unwritten.
Indeed, Shandy could describe the events of the nth day for the nth year and, thus, in his autobiography, every day would be captured.
In other words, if life lasted indefinitely, then it would have as many years as days.
Russell draws an analogy between this novel and Zeno with his tortoise. In his opinion, the solution lies in the fact that the whole is equivalent to its part at infinity. Those. only the “axiom of common sense” leads to a contradiction. However, the solution of the problem lies in the realm of pure mathematics. Obviously, there are two sets - years and days, between the elements of which there is a one-to-one correspondence - a bijection. Then, under the condition of the infinite life of the protagonist, there are two infinite sets of equal power, which, if we consider power as a generalization of the concept of the number of elements in a set, resolves the paradox.
Paradox (theorem) of Banach-Tarski or doubling the ball paradox- a theorem in set theory stating that a three-dimensional ball is equally composed of two of its copies.
Two subsets of the Euclidean space are called equally composed if one can be divided into a finite number of parts, moved them, and made up of them the second.
More precisely, two sets A and B are equally composed if they can be represented as a finite union of disjoint subsets such that for each i the subset is congruent.
If we use the choice theorem, then the definition sounds like this:
The axiom of choice implies that there is a partition of the surface of a unit sphere into a finite number of parts, which, by transformations of the three-dimensional Euclidean space that do not change the shape of these components, can be assembled into two spheres of unit radius.
Obviously, given the requirement for these parts to be measurable, this statement is not feasible. The famous physicist Richard Feynman in his biography told how at one time he managed to win the dispute about splitting an orange into a finite number of parts and recomposing it.
At certain points, this paradox is used to refute the axiom of choice, but the problem is that what we consider elementary geometry is not essential. Those concepts that we consider intuitive should be extended to the level of properties of transcendental functions.
To further weaken the confidence of those who believe that the axiom of choice is wrong, one should mention the theorem of Mazurkiewicz and Sierpinski, which states that there is a non-empty subset E of the Euclidean plane that has two disjoint subsets, each of which can be divided into a finite number of parts, so that they can be translated by isometries into a covering of the set E.
The proof does not require the use of the axiom of choice.
Further constructions based on the axiom of certainty give a resolution to the Banach-Tarski paradox, but are not of such interest.
- Richard's paradox: it is required to name "the smallest number not named in this book." The contradiction is that on the one hand, this can be done, since there is the smallest number named in this book. Proceeding from it, one can also name the smallest unnamed. But here a problem arises: the continuum is uncountable, between any two numbers you can insert an infinite number of intermediate numbers. On the other hand, if we could name this number, it would automatically move from the class not mentioned in the book to the class mentioned.
- The Grelling-Nilson paradox: words or signs can denote a property and at the same time have it or not. The most trivial formulation sounds like this: is the word “heterological” (which means “not applicable to itself”) heterological?.. It is very similar to Russell’s paradox due to the presence of a dialectical contradiction: the duality of form and content is violated. In the case of words that have a high level of abstraction, it is impossible to decide whether these words are heterological.
- Skolem's paradox: using Gödel's completeness theorem and the Löwenheim-Skolem theorem, we obtain that axiomatic set theory remains true even when only a countable set of sets is assumed (available) for its interpretation. In the same time
axiomatic theory includes the already mentioned Cantor's theorem, which brings us to uncountable infinite sets.
Resolution of paradoxes
The creation of set theory gave rise to what is considered the third crisis of mathematics, which has not yet been resolved satisfactorily for everyone.Historically, the first approach was set-theoretic. It was based on the use of actual infinity, when it was considered that any infinite sequence is completed in infinity. The idea was that in set theory one often had to operate on sets that could be parts of other, larger sets. Successful actions in this case were possible only in one case: the given sets (finite and infinite) are completed. A certain success was evident: Zermelo-Fraenkel's axiomatic set theory, a whole school of mathematics by Nicolas Bourbaki, which has existed for more than half a century and still causes a lot of criticism.
Logicism was an attempt to reduce all known mathematics to the terms of arithmetic, and then to reduce the terms of arithmetic to the concepts of mathematical logic. Frege took up this closely, but after finishing work on the work, he was forced to point out his inconsistency, after Russell pointed out the contradictions in the theory. The same Russell, as mentioned earlier, tried to eliminate the use of impredicative definitions with the help of "type theory". However, his concepts of set and infinity, as well as the axiom of reducibility, turned out to be illogical. The main problem was that the qualitative differences between formal and mathematical logic were not taken into account, as well as the presence of superfluous concepts, including those of an intuitive nature.
As a result, the theory of logicism could not eliminate the dialectical contradictions of the paradoxes associated with infinity. There were only principles and methods that made it possible to get rid of at least non-predicative definitions. In his own reasoning, Russell was Cantor's heir.
At the end of XIX - beginning of XX century. the spread of the formalist point of view on mathematics was associated with the development of the axiomatic method and the program of substantiation of mathematics, which was put forward by D. Hilbert. The importance of this fact is indicated by the fact that the first of the twenty-three problems he presented to the mathematical community was the problem of infinity. Formalization was necessary to prove the consistency of classical mathematics, "while excluding all metaphysics from it." Given the means and methods used by Hilbert, his goal turned out to be fundamentally impossible, but his program had a huge impact on the entire subsequent development of the foundations of mathematics. Hilbert worked on this problem for a long time, having first constructed the axiomatics of geometry. Since the solution of the problem turned out to be quite successful, he decided to apply the axiomatic method to the theory of natural numbers. Here is what he wrote in connection with this: “I pursue an important goal: it is I who would like to deal with the questions of the foundation of mathematics as such, turning every mathematical statement into a strictly derivable formula.” At the same time, it was planned to get rid of infinity by reducing it to a certain finite number of operations. To do this, he turned to physics with its atomism, in order to show the whole inconsistency of infinite quantities. In fact, Hilbert raised the question of the relationship between theory and objective reality.
A more or less complete idea of finite methods is given by Hilbert's student J. Herbran. By finite reasoning, he understands such reasoning that satisfies the following conditions: logical paradoxes "- only a finite and definite number of objects and functions are always considered;
Functions have a precise definition, and this definition allows us to calculate their value;
It never asserts "This object exists" unless a way to construct it is known;
The set of all objects X of any infinite collection is never considered;
If it is known that some reasoning or theorem is true for all these X, then this means that this general reasoning can be repeated for each specific X, and this general reasoning itself should be considered only as a model for such specific reasoning.
However, at the time of the last publication in this area, Gödel had already received his results, in essence he again discovered and approved the presence of dialectics in the process of cognition. In essence, the further development of mathematics demonstrated the failure of Hilbert's program.
What exactly did Gödel prove? There are three main results:
1. Gödel showed the impossibility of a mathematical proof of the consistency of any system large enough to include all arithmetic, a proof that would not use any other rules of inference than those found in the system itself. Such a proof, which uses a more powerful inference rule, may be useful. But if these rules of inference are stronger than the logical means of arithmetic calculus, then there will be no confidence in the consistency of the assumptions used in the proof. In any case, if the methods used are not finitist, then Hilbert's program will turn out to be impracticable. Gödel just shows the inconsistency of calculations for finding a finitist proof of the consistency of arithmetic.
2. Gödel pointed out the fundamental limitations of the possibilities of the axiomatic method: the Principia Mathematica system, like any other system with which arithmetic is built, is essentially incomplete, i.e. for any consistent system of arithmetic axioms there are true arithmetic sentences that are not derived from the axioms this system.
3. Gödel's theorem shows that no extension of an arithmetic system can make it complete, and even if we fill it with an infinite set of axioms, then in the new system there will always be true, but not deducible by means of this system, positions. The axiomatic approach to the arithmetic of natural numbers cannot cover the entire realm of true arithmetical propositions, and what we mean by the process of mathematical proof is not limited to the use of the axiomatic method. After Godel's theorem, it became meaningless to expect that the concept of a convincing mathematical proof could be given once and for all delineated forms.
The latest in this series of attempts to explain set theory was intuitionism.
He went through a number of stages in his evolution - semi-intuitionism, intuitionism proper, ultra-intuitionism. At different stages, mathematicians were worried about different problems, but one of the main problems of mathematics is the problem of infinity. The mathematical concepts of infinity and continuity have served as the subject of philosophical analysis since their inception (ideas of atomists, aporias of Zeno of Elea, infinitesimal methods in antiquity, infinitesimal calculus in modern times, etc.). The greatest controversy was caused by the use of various types of infinity (potential, actual) as mathematical objects and their interpretation. All these problems, in our opinion, were generated by a deeper problem - the role of the subject in scientific knowledge. The fact is that the state of crisis in mathematics is generated by the epistemological uncertainty of the comparison of the world of the object (infinity) and the world of the subject. The mathematician as a subject has the possibility of choosing the means of cognition - either potential or actual infinity. The use of potential infinity as a becoming, gives him the opportunity to carry out, to construct an infinite set of constructions that can be built on top of finite ones, without having a finite step, without completing the construction, it is only possible. The use of actual infinity gives him the opportunity to work with infinity as already realizable, completed in its construction, as actually given at the same time.
At the stage of semi-intuitionism, the problem of infinity was not yet independent, but was woven into the problem of constructing mathematical objects and ways to justify it. The semi-intuitionism of A. Poincaré and the representatives of the Parisian school of the theory of functions Baire, Lebesgue and Borel was directed against the acceptance of the axiom of free choice, with the help of which Zermelo's theorem is proved, which states that any set can be made completely ordered, but without indicating a theoretical way to determine the elements of any subset of the required sets. There is no way to construct a mathematical object, and there is no mathematical object itself. Mathematicians believed that the presence or absence of a theoretical method for constructing a sequence of objects of study can serve as the basis for substantiating or refuting this axiom. In the Russian version, the semi-intuitionistic concept in the philosophical foundations of mathematics was developed in such a direction as the effectivism developed by N.N. Luzin. Effectiveism is an opposition to the main abstractions of Cantor's doctrine of the infinite - actuality, choice, transfinite induction, etc.
For effectivism, the abstraction of potential feasibility is epistemologically more valuable than the abstraction of actual infinity. Thanks to this, it becomes possible to introduce the concept of transfinite ordinals (infinite ordinal numbers) on the basis of the effective concept of the growth of functions. The epistemological setting of effectiveism for displaying the continuous (continuum) was based on discrete means (arithmetic) and the descriptive theory of sets (functions) created by N.N. Luzin. The intuitionism of the Dutchman L. E. Ya. Brouwer, G. Weyl, A. Geyting sees freely emerging sequences of various types as a traditional object of study. At this stage, solving mathematical problems proper, including the restructuring of all mathematics on a new basis, intuitionists raised the philosophical question of the role of a mathematician as a cognizing subject. What is his position, where he is more free and active in choosing the means of cognition? Intuitionists were the first (and at the stage of semi-intuitionism) to criticize the concept of actual infinity, Cantor's theory of sets, seeing in it the infringement of the subject's ability to influence the process of scientific search for a solution to a constructive problem. In the case of using potential infinity, the subject does not deceive himself, since for him the idea of potential infinity is intuitively much clearer than the idea of actual infinity. For an intuitionist, an object is considered to exist if it is given directly to a mathematician or if the method of constructing it is known. In any case, the subject can begin the process of completing the construction of a number of elements of his set. The unconstructed object does not exist for intuitionists. At the same time, the subject working with actual infinity will be deprived of this opportunity and will feel the double vulnerability of the adopted position:
1) it is never possible to carry out this infinite construction;
2) he decides to operate with actual infinity as with a finite object, and in this case loses his specificity of the concept of infinity. Intuitionism consciously limits the possibilities of a mathematician by the fact that he can construct mathematical objects exclusively by means that, although obtained with the help of abstract concepts, are effective, convincing, provable, functionally constructive precisely practically and are themselves intuitively clear as constructions, constructions, the reliability of which in practice, there is no doubt. Intuitionism, relying on the concept of potential infinity and constructive research methods, deals with the mathematics of becoming, set theory refers to the mathematics of being.
For the intuitionist Brouwer, as a representative of mathematical empiricism, logic is secondary; he criticizes it and the law of the excluded middle.
In his partly mystical works, he does not deny the existence of infinity, but does not allow its actualization, only potentialization. The main thing for him is the interpretation and justification of practically used logical means and mathematical reasoning. The restriction adopted by intuitionists overcomes the uncertainty of the use of the concept of infinity in mathematics and expresses the desire to overcome the crisis in the foundation of mathematics.
Ultra-intuitionism (A.N. Kolmogorov, A.A. Markov and others) is the last stage in the development of intuitionism, at which its main ideas are modernized, significantly supplemented and transformed, without changing its essence, but overcoming shortcomings and strengthening positive aspects, guided by the criteria mathematical rigor. The weakness of the intuitionist approach was a narrow understanding of the role of intuition as the only source of justification for the correctness and effectiveness of mathematical methods. Taking "intuitive clarity" as a criterion of truth in mathematics, intuitionists methodologically impoverished the possibilities of a mathematician as a subject of knowledge, reduced his activity only to mental operations based on intuition and did not include practice in the process of mathematical knowledge. The ultra-intuitionistic program of substantiating mathematics is a Russian priority. Therefore, domestic mathematicians, overcoming the limitations of intuitionism, accepted the effective methodology of materialistic dialectics, recognizing human practice as a source of formation of both mathematical concepts and mathematical methods (inferences, constructions). The problem of the existence of mathematical objects was solved by ultraintuitionists, relying not on the undefined subjective concept of intuition, but on mathematical practice and a specific mechanism for constructing a mathematical object - an algorithm expressed by a computable, recursive function.
Ultra-intuitionism enhances the advantages of intuitionism, which consist in the possibility of ordering and generalizing the methods for solving constructive problems used by mathematicians of any direction. Therefore, intuitionism of the last stage (ultraintuitionism) is close to constructivism in mathematics. In the epistemological aspect, the main ideas and principles of ultraintuitionism are as follows: criticism of the classical axiomatics of logic; the use and significant strengthening (on the explicit instructions of A.A. Markov) of the role of abstraction of identification (mental abstraction from the dissimilar properties of objects and the simultaneous isolation of the general properties of objects) as a way of constructing and constructively understanding abstract concepts, mathematical judgments; proof of the consistency of consistent theories. In the formal aspect, the application of the abstraction of identification is justified by its three properties (axioms) of equality - reflexivity, transitivity and symmetry.
To solve the main contradiction in mathematics on the problem of infinity, which gave rise to a crisis of its foundations, at the stage of ultra-intuitionism in the works of A.N. Kolmogorov suggested ways out of the crisis by solving the problem of relations between classical and intuitionistic logic, classical and intuitionistic mathematics. Brouwer's intuitionism as a whole denied logic, but since any mathematician cannot do without logic, the practice of logical reasoning was still preserved in intuitionism, some principles of classical logic were allowed, having axiomatics as its basis. S.K. Kleene, R. Wesley even note that intuitionistic mathematics can be described as a kind of calculus, and calculus is a way of organizing mathematical knowledge on the basis of logic, formalization and its form - algorithmization. A new version of the relationship between logic and mathematics within the framework of intuitionistic requirements for intuitive clarity of judgments, especially those that included negation, A.N. Kolmogorov proposed as follows: he presented intuitionistic logic, closely related to intuitionistic mathematics, in the form of an axiomatic implicative minimal calculus of propositions and predicates. Thus, the scientist presented a new model of mathematical knowledge, overcoming the limitations of intuitionism in recognizing only intuition as a means of cognition and the limitations of logicism, which absolutizes the possibilities of logic in mathematics. This position made it possible to demonstrate in mathematical form the synthesis of the intuitive and the logical as the basis of flexible rationality and its constructive effectiveness.
Conclusions. Thus, the epistemological aspect of mathematical knowledge allows us to evaluate the revolutionary changes at the stage of the crisis of the foundations of mathematics at the turn of the 19th-20th centuries. from new positions in understanding the process of cognition, the nature and role of the subject in it. The epistemological subject of the traditional theory of knowledge, corresponding to the period of domination of the set-theoretic approach in mathematics, is an abstract, incomplete, “partial” subject, represented in subject-object relations, torn off by abstractions, logic, formalism from reality, rationally, theoretically knowing its object and understood as a mirror, accurately reflecting and copying reality. In fact, the subject was excluded from cognition as a real process and result of interaction with the object. The entry of intuitionism into the arena of the struggle of philosophical trends in mathematics led to a new understanding of the mathematician as a subject of knowledge - a person who knows, whose philosophical abstraction must be built, as it were, anew. The mathematician appeared as an empirical subject, already understood as an integral real person, including all those properties that were abstracted from in the epistemological subject - empirical concreteness, variability, historicity; it is an acting and cognizing in real cognition, a creative, intuitive, inventive subject. The philosophy of intuitionistic mathematics has become the basis, the foundation of the modern epistemological paradigm, built on the concept of flexible rationality, in which a person is an integral (holistic) subject of cognition, possessing new cognitive qualities, methods, procedures; he synthesizes his abstract-epistemological and logical-methodological nature and form, and at the same time receives an existential-anthropological and "historical-metaphysical" comprehension.
An important point is also intuition in cognition and, in particular, in the formation of mathematical concepts. Again, there is a struggle with philosophy, attempts to exclude the law of the excluded middle, as having no meaning in mathematics and coming into it from philosophy. However, the presence of an excessive emphasis on intuition and the lack of clear mathematical justifications did not allow transferring mathematics to a solid foundation.
However, after the emergence of a rigorous concept of an algorithm in the 1930s, the baton from intuitionism was taken over by mathematical constructivism, whose representatives made a significant contribution to the modern theory of computability. In addition, in the 1970s and 1980s, significant connections were discovered between some of the ideas of the intuitionists (even those that previously seemed absurd) and the mathematical theory of topos. The mathematics found in some topoi is very similar to that which the intuitionists were trying to create.
As a result, one can make a statement: most of the above paradoxes simply do not exist in the theory of sets with self-ownership. Whether such an approach is definitive is debatable, further work in this area will show.
Conclusion
Dialectical-materialistic analysis shows that paradoxes are a consequence of the dichotomy of language and thinking, an expression of deep dialectical (Gödel's theorem made it possible to manifest dialectics in the process of cognition) and epistemological difficulties associated with the concepts of an object and subject area in formal logic, a set (class) in logic and set theory, with the use of the principle of abstraction, which allows introducing new (abstract) objects (infinity), with methods for defining abstract objects in science, etc. Therefore, a universal way to eliminate all paradoxes cannot be given.Whether the third crisis of mathematics is over (because it was in a causal relationship with paradoxes; now paradoxes are an integral part) - opinions differ here, although formally known paradoxes were eliminated by 1907. However, now in mathematics there are other circumstances that can be considered either crisis or foreshadowing a crisis (for example, the absence of a rigorous justification for the path integral).
As for paradoxes, the well-known liar paradox played a very important role in mathematics, as well as a whole series of paradoxes in the so-called naive (preceding axiomatic) set theory that caused a crisis of foundations (one of these paradoxes played a fatal role in the life of H. Frege) . But, perhaps, one of the most underestimated phenomena in modern mathematics, which can be called both paradoxical and crisis, is Paul Cohen's solution in 1963 of Hilbert's first problem. More precisely, not the very fact of the decision, but the nature of this decision.
Literature
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I. Basic concepts and axioms of set theory
Over the thousands of years of its existence, from the simplest ideas about the number and figure, mathematics has come to the formation of many new concepts and methods. It has become a powerful tool for the study of nature and a flexible instrument of practice. The 20th century brought new ideas and theories to mathematics, and the scope of its application expanded. Mathematics occupies a special position in the system of sciences - it cannot be attributed to either the humanities or the natural sciences. But she introduced the basic concepts that are used in them. Such a concept is the concept of "set", which first arose in mathematics and is now general scientific.
The first draft of set theory is by Bernard Bolzano (Paradoxes of the Infinite, 1850). In this work, arbitrary (numerical) sets are considered, and for their comparison, the concept of one-to-one correspondence is defined.
At the end of the 19th century, Georg Cantor, the German mathematician and founder of set theory, gave an intuitive definition of the concept of "set" as follows: "Many is many thought as a whole". Such a definition of a set required the introduction three characters.
First of them must represent the multitude as something “one”, i.e. be representative of the multitude. As such a symbol, it is customary to use any capital letter of any alphabet: for example, to designate sets with capital letters of the Latin alphabet A, B, ..., X, or any other by agreement.
Second the symbol must represent "many", that is, be considered as an element of a set. As this symbol, it is customary to use lowercase letters of the same alphabet: a, b, ..., z.
Third a symbol must unambiguously relate an element to a set. The sign is defined as the corresponding symbol, which comes from the first letter of the Greek word (to be). The entry defines the relationship: x is an element of X. To indicate that x is not an element of X, write .
It should be noted that such a definition of the concept of a set leads to a number of internal contradictions of the theory - the so-called paradoxes.
For example, consider Russell's paradox. Hairdresser
(element x) living in some village who do not shave themselves (let X be the set of all those and only those inhabitants of the given village who do not shave themselves). Does the barber shave himself? That is, or? It is impossible to answer the question, because assuming, for example, that , we immediately come to a contradiction: , and vice versa.
In the school course of mathematics, students consider the concept of a set as an indefinable concept, which is understood as a set of objects of the reality around us, conceived as a single whole. And each object of this collection is called element of this set.
Currently, there are several axiomatic systems of set theory:
Zermelo's system of axioms. This system of axioms is often supplemented with the axiom of choice, and is called the Zermelo-Fraenkel system with the axiom of choice (ZFC).
Axioms of the NBG theory. This system of axioms, proposed by von Neumann, was later revised and simplified by Robinson, Bernays and Gödel.
The Zermelo system (Z-system) consists of 7 axioms. Let us describe these axioms within the framework in which they are used in the school mathematics course.
Axiom of volume (Z1). If all elements of set A belong to set B, and all elements of set B also belong to set A, then A=B.
To clarify this axiom, we need to use the term "subset": If each element of the set A is an element of the set Z, then we say that A is subset Z, and write . The symbol is called "on". If the possibility of a situation when Z=A is not excluded, then in order to focus on this, they write.
Introducing the term "subset", we formulate axiom 1 in symbolic form: .
Pair axiom (Z2). For arbitrary a and b, there is a set whose only elements are (a,b).
This axiom is used to explain the Cartesian product of sets, where the initial concept is an "ordered pair". Under ordered pair understand the totality of two elements, each of which occupies a certain place in the record. An ordered pair is denoted as follows: (a, b).
The sum axiom (Z3). For arbitrary sets A and B, there is a unique set C whose elements are all elements of set A and all elements of set B and which no longer contains any other elements.
In symbolic form, axiom Z3 can be written as follows: . Based on this axiom and the theorems following from it, the properties of set operations are indicated, the description of which will be presented in Section 3. The axioms Z1 and Z2 allow us to introduce the concept of the operation of union, intersection, addition, difference of sets.
Axiom of degree (Z4). For any set X, there is a set of all its subsets P(X).
Axiom of infinity (Z6). There is at least one infinite set - the natural series of numbers.
Axiom of Choice (Z7). For any family of non-empty sets, there is a function that associates with each set of the family one of the elements of this set. The function is called selection function for a given family.
It is worth noting the importance of the corresponding axioms, since sets and relations between them are the subject of study of any mathematical discipline.
We point out another important discovery in set theory - the image of relations between subsets, for visual representation. One of the first to use this method was the outstanding German mathematician and philosopher Gottfried Wilhelm Leibniz. Then this method was developed quite thoroughly by Leonhard Euler. After Euler, the same method was developed by the Czech mathematician Bernard Bolzano. Only, unlike Euler, he did not draw circular, but rectangular diagrams. The Euler circle method was also used by the German mathematician Ernest Schroeder. But graphic methods reached their greatest flourishing in the writings of the English logician John Venn. In honor of Venn, instead of Euler circles, the corresponding figures are sometimes called Venn diagrams, and in some books they are also called Euler-Venn diagrams. Euler-Venn diagrams are used not only in mathematics and logic, but also in management and other applied areas.
II. Relations between sets and ways to define them
So, sets are understood as a set of any objects, conceivable as a single whole. Sets may consist of objects of very different natures. Their elements can be letters, atoms, numbers, equations, points, angles, etc. This explains the extreme breadth of set theory and its application to the most diverse fields of knowledge (mathematics, physics, economics, linguistics, etc.).
It is believed that a set is defined by its elements, that is, a set is given if any object can be said to belong to this set or not. There are two ways of specifying sets.
- element enumerations.
For example, if the set A consists of elements a, b, c, then they write: A = (a, b, c).
Not every set can be specified using an enumeration of elements. Sets all of whose elements can be enumerated are called finite. Sets all of whose elements cannot be enumerated are called infinite. They cannot be specified using an enumeration of elements. The exception is infinite sets, in which the order of formation of each next element on the basis of the previous one is clear. For example, the set of natural numbers is an infinite set. But it is known that in it each next number, starting from the second, is 1 more than the previous one. Therefore, you can set N = (1, 2, 3, 4, ...) as follows.
- The set can be specified using indication of a characteristic property.
characteristic property of a given set is a property that all elements of this set have and none of the elements that do not belong to it have. It is denoted: A = (x|…), where after the vertical bar the characteristic property of the elements of this set is written.
For example, B=(1,2,3). It is easy to see that each element of the set B is a natural number less than 4. It is this property of the elements of the set B that is characteristic for it. In this case, they write: and read: “The set B consists of elements x such that x belongs to the set of natural numbers and x is less than four” or the set B consists of natural numbers less than 4. The set B can also be specified in another way: or, etc.
Moreover, if an element does not obey the characteristic property of the set, then it does not belong to this set. There are sets that can only be specified by specifying a characteristic property, for example, .
Of particular importance in the school course of mathematics are number sets, i.e. a set whose elements are numbers. For the name of numerical sets in mathematics, special notation is accepted:
N = (1, 2, 3, 4, …) - set of natural numbers;
Z = (…,-4, -3, -2, -1, 0, 1, 2, 3, 4, …) - a set of integers (contains all natural numbers and their opposites);
Q = (x | x=p/q, where p∈Z, q∈N) - the set of rational numbers (consists of numbers that can be represented as an ordinary fraction);
J - a set of irrational numbers (a set consisting of infinite decimal non-periodic fractions, for example: 1,23456342 …;, and etc.)
R = (-∞; +∞) - the set of real numbers.
The set of all real numbers L. Euler depicted using circles. (Fig. 1)
It should be noted that all any numerical sets can be specified using a numerical interval. (Fig. 2)
Types of numeric ranges
Set C, discussed above, is a numeric set and can be specified using a numeric gap (Fig. 3)
Figure 3 - Numeric gap
Let us point out one more important rule for specifying numerical sets: Finite numerical sets are depicted on the real line by separate points.
In mathematics, one sometimes has to consider sets containing only one element, and even sets that do not have a single element. A set that does not contain any element is called empty. It is denoted by the sign ∅. For example, given a set A=(x|x∈N∧-2
It is worth noting that when it comes to two or more sets, there may or may not be any relationship between them. If the sets are in any relationship, then we are talking about a relation equality or relation inclusion.
Set A turns on to set B, if each element of set A belongs to set B. This relation is denoted as follows: A⊂B. Or, in another way, they say that set A is a subset of set B.
The sets A and B are called equal, if and only if each element of set A belongs to set B and at the same time each element of set B belongs to set A. This relation is denoted as follows: A \u003d B
For example:
1) A=(a,b,c,d) and B=(b,d), these sets are in relation to the inclusion B⊂A, because Every element of set B belongs to set A.
2) M=(x|x∈R∧x<6}=(-∞;6) и K{x|x∈R∧x≤8}=(-∞;8], эти множества находятся в отношении включения M⊂K, т.к. каждый элемент множества M принадлежит множеству K (Рис. 4)
Figure 4 - Numeric gap
3) A=(x|x∈N∧x:2)=(2,4,6,8,10,...) and B=(x|x∈N∧x:3)=(3,6 ,9,12,...), these two sets are not in any relationship A⊄B, since the set A has an element 2 that does not belong to the set B
and B⊄A, because in set B there is an element 3 that does not belong to set A.
Therefore, these sets are not in any relationship.
III. Operations and properties of operations on sets
Def.1. crossing sets A and B is an operation, the result of which is a set consisting of those and only those elements that belong to both A and B at the same time.
A∩B=(x|x∈A∧x∈B)
Def.2.Association sets A and B is an operation, the result of which is a set consisting of those and only those elements that belong to the set A or set B (ie, at least one of these sets).
A∪B=(x|x∈A∨x∈B)
Def.3. difference sets A and B is called an operation, the result of which is a set consisting of those and only those elements that belong to A and not belong to B at the same time.
A\ B =(x∈A∧x∉B)
Def.4. Complementing the set A to the universal set A set is called a set, each element of which belongs to the universal and does not belong to A.
Set expressions
From sets, signs of operations on them and, perhaps, brackets, expressions can be formed. For example, A∩B\C.
You need to know the order of operations in such expressions and be able to read them.
Order of Operations
if there are no brackets, then first of all, the addition to the universal set of a simple set is performed, then the intersection and union (they are equal to each other), and lastly, the difference;
if the expression contains brackets, then first perform the operations in the brackets in the order given in paragraph 1), and then all the operations outside the brackets.
For example, a) A∩B\C; b) A∩(B\C); c) A∩(B\C)" .
The reading of the expression starts from the result of the last operation. For example, expression a) is read as follows: the difference of two sets, the first of which is the intersection of sets A and B, and the second is the set C.
Euler circles
Operations on sets and relations between them can be represented using Euler circles. These are special drawings in which ordinary sets are represented by circles, the universal set by a rectangle.
Task. Draw the set (A∪B)"∩C using Euler circles.
Solution. Let's arrange the order of execution of operations in this expression: (A∪B) "∩C. Shade the results of operations according to the order of their execution
Set operation properties(fig.5)
Properties I - 8 and 1 0 - 8 0 are interconnected by the so-called principle of duality:
if in any of the two columns of properties the signs ∩→∪, ∪→∩, ∅→U, U→∅ are reversed, then another column of properties will be obtained.
IV. Partitioning a set into classes
It is considered that the set X is divided into pairwise disjoint subsets or classes if the following conditions are met:
1) the intersection of any two subsets is empty;
2) the union of all subsets coincides with the set X.
The division of a set into classes is called a classification.
V. Cartesian product of sets
The Cartesian product of sets A and B is a set of pairs, the first component of each of which belongs to set A, and the second to set B. The Cartesian product of sets A and B is denoted by A x B. Thus, A×B=((x,y)|x ∈A˄y∈B). The operation of finding the Cartesian product of sets A and B is called the Cartesian multiplication of these sets. If A and B are numerical sets, then the elements of the Cartesian product of these sets will be ordered pairs of numbers.
VI. Sum and product rules
Denote the number of elements of a finite set A by n(A). If sets A and B do not intersect, then n(AUB)= n(A) + n(B). If sets A and B intersect, then n(A U B) = n (A) + n (B) - n (A ∩ B).
The number of elements of the Cartesian product of sets A and B is calculated by the formula n (A X B) = n (A) . n(B).
The rule for counting the number of elements of the union of disjoint finite sets in combinatorics is called the sum rule, if the element x can be chosen in k ways, and the element y in m ways, and none of the ways to choose the element x coincides with the way to choose the element y, then the choice "x or y" can be done in k + m ways.
The rule for counting the number of elements of a Cartesian product of finite sets in combinatorics is called the product rule: if an element x can be chosen in k ways, and an element y in m ways, then the pair (x, y) can be chosen in km ways.
VII. List of sources used
Aseev G.G. Abramov O.M., Sitnikov D.E. Discrete Mathematics: Textbook. - Rostov n / a: "Phoenix", Kharkov: "Torsing", 2003, -144s.
Vilenkin N. Ya. Algebra. Textbook for IX - X grades of secondary schools with a mathematical specialization, 1968
Vilenkin N.Ya. Set stories. M.: Publishing house "Science". - 1965. - 128s
Euler diagrams - Venn.URL: http://studopedia.net/1_5573_diagrammi-eylera-venna.html
Kireenko S.G., Grinshpon I.E. Elements of set theory (textbook). - Tomsk, 2003. - 42 p.
Kuratovsky K., Mostovsky A. Theory of sets. - M.: Mir, 1970, - 416s.
