EntranceAn axiomatic method for constructing a natural series of numbers. Axiomatic construction of the set of natural numbers
Requirements for the axiom system, Peano axioms. When axiomatically constructing any mathematical theory, certain rules are observed: 1) some concepts of the theory are chosen as basic and accepted without definition; 2) each concept of the theory that is not contained in the list of basic ones is given a definition. It explains its meaning with the help of basic and preceding concepts. 3) axioms are formulated, that is, propositions that in a given theory are accepted without proof. Axioms reveal the properties of basic concepts. 4) every proposition of the theory that is not contained in the list of axioms must be proven. Such propositions are called theorems. They are proven on the basis of axioms and theorems preceding this one.
THAT. the axiomatic method of constructing a mathematical theory goes through several stages: 1) introduction of basic undefined concepts (eg: set, element of set in set theory). 2) introduction of basic relations (eg: membership relation in set theory). 3) by indicating the basic concepts and basic relations, the definition of other concepts and relations is introduced (for example: in set theory, the concepts of union, intersection, difference, complement).
In the axiomatic construction of a theory, all statements are derived by proof from axioms. The basis of such a theory is a system of axioms, and special requirements are imposed on the axiom system: 1) the axiom system must be consistent. A system of axioms is called consistent if two mutually exclusive propositions cannot be logically deduced from it. In other words, it is impossible to derive a statement and the negation of a given statement so that they are simultaneously true. To verify the consistency of the axiom system, it is enough to build a model of this system. 2) the system of axioms must be independent. A system of axioms is called independent if none of the axioms of this system are consequences of other axioms. In other words, each axiom of this system cannot be deduced from the other axioms. To prove the independence of a system of axioms, it is enough to build a model of this system. 3) the system of axioms must be complete, i.e. the number of axioms chosen in a given theory should be sufficient to introduce new concepts, relations, prove theorems, and to build the entire theory.
When constructing the same theory axiomatically, different systems of axioms can be used, but they must be equivalent. The relation “directly follow” is taken as the basic concept in the axiomatic construction of a system of natural numbers. The concepts of “set”, “element of a set”, and a rule of logic are also considered well-known. The element immediately following element a is designated a - prime.
The essence of the relationship “directly follow” is revealed in the following axioms: 1) in the set of natural numbers there is an element that does not directly follow any element of this set, this element 1 (one). 2) for each element a from the set of natural numbers (N) there is a unique element a? , immediately following a. 3) for each element a of N, there is at most one element immediately followed by a. 4) any subset M of the set N that has the properties: 1 M, and from the fact that a is contained in M, what is a? contained in M, coincides with the set N.
The listed systems of axioms are called Peano axioms. THAT. the set of numbers for which the immediately following relation is established, satisfying the Peano axioms, is called the set of natural numbers, and its element is called a natural number. The fourth axiom describes the infinity of the natural series of numbers and is called the axiom of induction. On its basis, a proof of various statements is carried out using the method of mathematical induction, which is as follows: in order to prove that a given statement is true for any natural number, it is necessary: 1) to prove that this statement is true for one, 2) from the proposition that the statement is true for an arbitrary number k, prove that it is true for the next number k?.
The definition of the set N says nothing about the nature of this set, which means it can be anything. Choosing as a set N any set on which the relation to immediately follow and satisfying the Peano axioms is given, we obtain a model of this axiom system. A one-to-one correspondence can be established between all such models. These models will differ only in the nature of the elements, name and designation. No.: 1, 2, 3, 4, 5… 0.00,000,0000,00000,… Ѕ, 1/3, ј, 1/5,
When axiomatically constructing any mathematical theory, certain rules:
· some concepts of the theory are chosen as basic and accepted without definition;
· each concept of the theory that is not contained in the list of basic ones is given a definition;
· axioms are formulated - propositions that in a given theory are accepted without proof; they reveal the properties of basic concepts;
· every proposition of the theory that is not contained in the list of axioms must be proven; Such propositions are called theorems and are proven on the basis of axioms and theorems.
In the axiomatic construction of a theory, all statements are derived from axioms through proof.
Therefore, special requirements apply to the system of axioms. requirements:
· consistency (a system of axioms is called consistent if two mutually exclusive propositions cannot be logically deduced from it);
· independence (a system of axioms is called independent if none of the axioms of this system is a consequence of other axioms).
A set with a relation specified in it is called a model of a given axiom system if all the axioms of the given system are satisfied in it.
There are many ways to construct a system of axioms for a set of natural numbers. For example, a sum of numbers or an order relation can be taken as a basic concept. In any case, you need to define a system of axioms that describe the properties of basic concepts.
Let us give a system of axioms, accepting the basic concept of the operation of addition.
Non-empty set N we call it a set of natural numbers if the operation is defined in it (a; b) → a + b, called addition and having the following properties:
1. addition is commutative, i.e. a + b = b + a.
2. addition is associative, i.e. (a + b) + c = a + (b + c).
4. in any set A, which is a subset of the set N, Where A there is a number and such that everything Ha, are equal a+b, Where bN.
Axioms 1 - 4 are enough to construct the entire arithmetic of natural numbers. But with such a construction it is no longer possible to rely on the properties of finite sets that are not reflected in these axioms.
Let us take as the basic concept the relation “directly follow...”, defined on a non-empty set N. Then the natural series of numbers will be the set N, in which the relation “immediately follow” is defined, and all elements of N will be called natural numbers, and the following holds: Peano's axioms:
AXIOM 1.
In abundanceNthere is an element that does not immediately follow any element of this set. We will call it unity and denote it by the symbol 1.
AXIOM 2.
For each element a ofNthere is a single element a immediately following a.
AXIOM 3.
For each element a ofNThere is at most one element immediately followed by a.
AXOIMA 4.
Any subset M of the setNcoincides withN, if it has the following properties: 1) 1 is contained in M; 2) from the fact that a is contained in M, it follows that a is also contained in M.
A bunch of N, for whose elements the relation “directly follow...” is established, satisfying axioms 1 - 4, is called set of natural numbers , and its elements are natural numbers.
If as a set N choose some specific set on which a specific relation “directly follow ...” is given, satisfying axioms 1 - 4, then we get different interpretations (models) given axiom systems.
The standard model of the Peano axiom system is a series of numbers that emerged in the process of historical development of society: 1, 2, 3, 4, 5, ...
The model of the Peano axioms can be any countable set.
For example, I, II, III, IIII, ...
oh oh oh oh oh...
one two three four, …
Let's consider a sequence of sets in which set (oo) is the initial element, and each subsequent set is obtained from the previous one by adding another circle (Fig. 15).
Then N there is a set consisting of sets of the described form, and it is a model of the Peano axiom system.
Indeed, in many N there is an element (oo) that does not immediately follow any element of the given set, i.e. Axiom 1 is satisfied. For each set A of the population under consideration there is a single set that is obtained from A by adding one circle, i.e. Axiom 2 holds. For each set A there is at most one set from which a set is formed A by adding one circle, i.e. Axiom 3 holds. If MN and it is known that many A contained in M, it follows that a set in which there is one more circle than in the set A, also contained in M, That M =N, and therefore axiom 4 is satisfied.
In the definition of a natural number, none of the axioms can be omitted.
Let us establish which of the sets shown in Fig. 16 are a model of the Peano axioms.
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Solution. Figure 16 a) shows a set in which axioms 2 and 3 are satisfied. Indeed, for each element there is a unique one immediately following it, and there is a unique element that it follows. But in this set, axiom 1 is not satisfied (axiom 4 does not make sense, since there is no element in the set that does not immediately follow any other). Therefore, this set is not a model of the Peano axioms.
Figure 16 b) shows a set in which axioms 1, 3 and 4 are satisfied, but behind the element A two elements immediately follow, and not one, as required in axiom 2. Therefore, this set is not a model of the Peano axioms.
In Fig. 16 c) shows a set in which axioms 1, 2, 4 are satisfied, but the element With immediately follows two elements immediately. Therefore, this set is not a model of the Peano axioms.
In Fig. 16 d) shows a set that satisfies axioms 2, 3, and if we take the number 5 as the initial element, then this set will satisfy axioms 1 and 4. That is, in this set for each element there is a unique one immediately following it, and there is a single element that it follows. There is also an element that does not immediately follow any element of this set, this is 5 , those. Axiom 1 is satisfied. Accordingly, Axiom 4 will also be satisfied. Therefore, this set is a model of Peano’s axioms.
Using Peano's axioms, we can prove a number of statements. For example, we will prove that for all natural numbers the inequality x x.
Proof. Let us denote by A set of natural numbers for which a a. Number 1 belongs A, since it does not follow any number from N, which means it does not follow by itself: 1 1. Let aA, Then a a. Let's denote A through b. By virtue of axiom 3, Ab, those. b b And bA.
OZO MATHEMATICS 1st year 2nd semester
Example 1: Let us justify the choice of action when solving the problem: “We bought 4 packs of colored paper, and 3 packs more white paper. How many packs of white paper did you buy?
Solution. The problem deals with two sets. Let A be a set of packs of colored paper, B be a set of packs of white paper. By condition, the number of packs of colored paper is known, i.e. n(A)=4, and the size of set B needs to be found. In addition, according to the conditions of the problem, in set B we can select a subset C, the number of which is 3, i.e. n(C)=3. Let's do this, for example, as shown in Fig. 1.
Picture 1
Then the difference B \ C = B 1 will be equal to the set A, i.e. n(B 1) = n(A).
Thus, set B is the union of sets B 1 and C, where B 1 C=Æ.
The problem comes down to determining the size of the union of two disjoint sets and is solved by addition: n(B) = n(B 1 C) = n(B 1) + n(C); n(B) = 4+3 = 7.
Example 2: Using the concept of number as a measure of magnitude, we will justify the choice of action when solving the problem: “3m of fabric was used for the skirt, and 2m for the blouse. How many meters of fabric went into the whole suit?
Solution: The problem deals with a quantity - length, which is measured using a unit of 1 meter, because the length is continuous, then we will explain the choice of action when solving the problem using segments (Fig. 2).
Let e=1m, segment a shows the length of the fabric used for the skirt, a=3e. The segment b shows the length of the fabric used for the blouse, b = 2e. Because In the problem you need to find out the amount of all the fabric used, then the segment c will indicate the amount of all the fabric used: c = a + b.
 Figure 2 | a=3e b=2e m e (c)= m e (a)+m e (c) m e (c) = 2+3 m e (c) = 5 Answer: 5 m. |
Example 3: Using the concept of number as a measure of magnitude, we will justify the choice of action when solving the problem: “In the first box there were 12 kg of cookies, and in the second there were 3 kg less. How many kilograms of cookies were in the second box?
Solution: The problem deals with the quantity mass, the unit of measurement of which is 1 kilogram, e = 1 kg, because quantity, the mass is continuous, then we will explain the choice of action when solving the problem using segments (Fig. 3).
Let e = 1kg, segment a shows how many kilograms of cookies were in the first box, a = 12e.
Segment b shows how many kilograms of cookies were in the second box less than in the first, b = 3e.
The segment c shows how many kilograms of cookies were in the second box, m e (c) - ? It is known that the second box contains 3 kg of cookies less than the first, i.e. the same, but 3 less.
| Let d=a, then c = d – b. a = 12e, which means d = 12e. m e (c)= m e (d)-m e (c) m e (c)=12-3 m e (c)=9 |  Figure 3
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Answer: There were 9 kilograms of cookies in the second box.
When axiomatically constructing any mathematical theory, certain rules are observed:
Some concepts of the theory are chosen as main and are accepted without definition;
Each concept of the theory that is not contained in the list of basic ones is given a definition, its meaning is explained in it with the help of basic and previous concepts;
Are formulated axioms- proposals that are accepted without proof in this theory; they reveal the properties of basic concepts;
Every proposition of a theory that is not contained in the list of axioms must be proven; such propositions are called theorems and are proven on the basis of axioms and theorems preceding the one under consideration.
If the construction of a theory is carried out using the axiomatic method, i.e. according to the rules mentioned above, then they say that the theory is constructed deductively.
In the axiomatic construction of a theory, essentially all statements are derived by proof from axioms. Therefore, special requirements are placed on the axiom system. First of all, it must be consistent and independent.
The system of axioms is called consistent, if two mutually exclusive sentences cannot be logically deduced from it.
If a system of axioms does not have this property, it cannot be suitable for substantiating a scientific theory.
A consistent system of axioms is called independent, if none of the axioms of this system is a consequence of other axioms of this system.
When constructing the same theory axiomatically, different systems of axioms can be used. But they must be equivalent. In addition, when choosing a particular system of axioms, mathematicians take into account how simply and clearly proofs of the theorems can be obtained in the future. But if the choice of axioms is conditional, then science itself or a separate theory does not depend on any conditions - they are a reflection of the real world.
The axiomatic construction of a system of natural numbers is carried out according to the formulated rules. By studying this material, we must see how the entire arithmetic of natural numbers can be derived from basic concepts and axioms. Of course, its presentation in our course will not always be strict - we omit some proofs due to their great complexity, but we will discuss each such case.
Exercise
1. What is the essence of the axiomatic method of constructing a theory?
2. Is it true that an axiom is a proposition that does not require proof?
3. Name the basic concepts of the school planimetry course. Remember a few axioms from this course. The properties of what concepts are described in them?
4. Define a rectangle, choosing a parallelogram as a generic concept. Name three concepts that should precede the concept of “parallelogram” in a geometry course.
5. What sentences are called theorems? Remember what the logical structure of the theorem is and what it means to prove the theorem.
Basic concepts and axioms. Definition of natural number
As the basic concept in the axiomatic construction of the arithmetic of natural numbers, the relation “directly follow” is taken, defined on a non-empty set N. The concept of a set, an element of a set and other set-theoretic concepts, as well as the rules of logic, are also considered well-known.
The element immediately following the element A, denote A".
The essence of the “directly follow” attitude is revealed in the following axioms.
Axiom 1. In the set N there is an element that does not immediately follow any element of this set. We will call it unity and denote it by the symbol 1.
Axiom 2. For each element and from N there is only one element a", immediately following A.
Axiom 3. For each element A There is at most one element in N that is immediately followed by A.
Axiom 4. Every subset M sets N coincides with N, if it has the following properties: 1) 1 is contained in M; 2) from the fact that A contained in M, it follows that A" contained in M.
The formulated axioms are often called Peano axioms.
Using the "immediately follow" relation and axioms 1-4, we can give the following definition of a natural number.
Definition. A bunch of N, for whose elements the relation “directly follow” is established, satisfying axioms 1-4, is called a set of natural numbers, and its elements- natural numbers.
This definition says nothing about the nature of the elements of the set N. So it can be anything. Choosing as
set N is some specific set on which a specific “directly follow” relation is specified, satisfying axioms 1-4, we get model of a given system of axioms. It has been proven in mathematics that a one-to-one correspondence can be established between all such models, preserving the “directly follow” relation, and all such models will differ only in the nature of the elements, their name and designation. The standard model of the Peano axiom system is a series of numbers that emerged in the process of historical development of society:
Each number in this series has its own designation and name, which we will consider known.
Considering the natural series of numbers as one of the models of axioms 1-4, it should be noted that they describe the process of formation of this series, and this happens when the properties of the relation “directly follow” are revealed in the axioms. Thus, the natural series begins with the number 1 (axiom 1); every natural number is immediately followed by a single natural number (axiom 2); every natural number immediately follows at most one natural number (axiom 3); starting from the number 1 and moving in order to the natural numbers immediately following each other, we obtain the entire set of these numbers (axiom 4). Note that axiom 4 formally describes the infinity of the natural series, and the proof of statements about natural numbers is based on it.
In general, the model of the Peano axiom system can be any countable set, for example:!..
Consider, for example, a sequence of sets in which set (oo) is the initial element, and each subsequent set is obtained from the previous one by adding another circle (Fig. 108, a). Then N there is a set consisting of sets of the described form, and it is a model of the Peano axiom system. Indeed, in the set N there is an element (oo) that does not immediately follow any element of this set, i.e.
there is a unique set that can be obtained from A by adding one circle, i.e., axiom 2 is satisfied. For each set A there is at most one set from which a set is formed A by adding one circle, i.e. Axiom 3 holds. If MÌ N and it is known that many A contained in M, it follows that a set in which there is one more circle than in the set A, also contained in M, That M = N(and therefore, axiom 4 is satisfied).
Note that in the definition of a natural number, none of the axioms can be omitted - for any of them it is possible to construct a set in which the other three axioms are satisfied, but this axiom is not satisfied. This position is clearly confirmed by the examples given in Figures 109 and 110. Figure 109a shows a set in which axioms 2 and 3 are satisfied, but axiom 1 is not satisfied (axiom 4 will not make sense, since there is no element in the set, directly not following any other). Figure 109b shows a set in which axioms 1, 3 and 4 are satisfied, but behind the element A two elements immediately follow, and not one, as required in axiom 2. Figure 109c shows a set in which axioms 1, 2, 4 are satisfied, but the element With immediately follows as element A, and behind the element b. Figure 110 shows a set in which axioms 1, 2, 3 are satisfied, but axiom 4 is not satisfied - a set of points lying on the ray, it contains the number immediately following it, but it does not coincide with the entire set of points shown in the figure.
The fact that axiomatic theories do not talk about the “true” nature of the concepts being studied makes these theories too abstract and formal at first glance - it turns out that the same axioms are satisfied by different sets of objects and different relations between them. However, this apparent abstraction is the strength of the axiomatic method: every statement derived logically from these axioms is applicable to any sets of objects, as long as relations that satisfy the axioms are defined in them.
So, we began the axiomatic construction of a system of natural numbers by choosing the basic relation “immediately follow” and the axioms that describe its properties. Further construction of the theory involves consideration of the known properties of natural numbers and operations on them. They must be disclosed in definitions and theorems, i.e. are derived purely logically from the relation “directly follow”, and axioms 1-4.
The first concept we will introduce after defining the natural number is the “immediately precedes” relation, which is often used when considering the properties of the natural number.
Definition. If a natural number b immediately follows a natural number a, then the number a is said to immediately precede (or precede) the number b.
The relation “precedes” has a number of properties. They are formulated as theorems and proven using axioms 1 – 4.
Theorem 1. The unit has no preceding natural number.
The truth of this statement follows immediately from axiom 1.
Theorem 2. Every natural number A, different from 1, has a preceding number b, such that b ¢ = a.
Proof. Let us denote by M the set of natural numbers consisting of the number 1 and all numbers that have a predecessor. If the number A contained in M, that's the number A" also available in M, since it precedes for A" is the number A. This means that many M contains 1, and from the fact that the number A belongs to the set M, it follows that the number A" belongs M. Then, by axiom 4, the set M coincides with the set of all natural numbers. This means that all natural numbers except 1 have a preceding number.
Note that by virtue of Axiom 3, numbers other than 1 have a single preceding number.
The axiomatic construction of the theory of natural numbers is not considered either in primary or secondary schools. However, those properties of the relation “directly follow”, which are reflected in Peano’s axioms, are the subject of study in the initial course of mathematics. Already in the first grade, when considering the numbers of the first ten, it becomes clear how each number can be obtained. The concepts “follows” and “precedes” are used. Each new number acts as a continuation of the studied segment of the natural series of numbers. Students are convinced that each number is followed by the next, and, moreover, only one thing, that the natural series of numbers is infinite. And of course, knowledge of axiomatic theory will help the teacher methodically and competently organize children’s assimilation of the features of the natural series of numbers.
Exercises
1.Can axiom 3 be formulated as follows: “For each element A from N there is a single element that is immediately followed by a"?
2. Select the condition and conclusion in axiom 4, write them using the symbols О, =>.
3.Continue the definition of a natural number: “A natural number is an element of the set Î, Þ.
Addition
According to the rules for constructing an axiomatic theory, the definition of addition of natural numbers must be introduced using only the relation “immediately follow”, and the concepts “natural number” and “preceding number”.
Let us preface the definition of addition with the following considerations. If to any natural number A add 1, we get the number A", immediately following a, i.e. A + 1 = A", and, therefore, we get the rule for adding 1 to any natural number. But how to add to a number A natural number b, different from 1? Let's use the following fact: if we know that 2 + 3 = 5, then the sum 2 + 4 is equal to the number 6, which immediately follows the number 5. This happens because in the sum 2 + 4 the second term is the number immediately following the number 3 Thus, the amount A+ b" can be found if the amount is known A+ b. These facts form the basis for the definition of addition of natural numbers in axiomatic theory. In addition, it uses the concept of algebraic operation.
Definition. The addition of natural numbers is an algebraic operation that has the following properties:
1) ("A Î N ) a + 1=a",
2) (" A, b Î) a + b" = (a + b)".
Number A+ b called the sum of numbers A And b, and the numbers themselves A And b-terms.
As is known, the sum of any two natural numbers is also a natural number, and for any natural numbers A And b sum A+ b- the only one. In other words, the sum of natural numbers exists and is unique. The peculiarity of the definition is that it is not known in advance whether there is an algebraic operation that has the specified properties, and if it does exist, is it unique? Therefore, when constructing the axiomatic theory of natural numbers, the following statements are proven:
Theorem 3. Addition of natural numbers exists and it is unique.
This theorem consists of two statements (two theorems):
1) addition of natural numbers exists;
2) addition of natural numbers is unique.
As a rule, existence and uniqueness are linked together, but they are most often independent of each other. The existence of an object does not imply its uniqueness. (For example, if you say that you have a pencil, this does not mean that there is only one.) A uniqueness statement means that there cannot be two objects with given properties. Uniqueness is often proven by contradiction: one assumes that there are two objects that satisfy a given condition, and then builds a chain of deductive inferences that leads to a contradiction.
To verify the truth of Theorem 3, we first prove that if in the set N there is an operation with properties 1 and 2, then this operation is unique; then we will prove that the operation of addition with properties 1 and 2 exists.
Proof of the uniqueness of addition. Let us assume that in the set N There are two addition operations that have properties 1 and 2. We denote one of them by the + sign, and the other by the Å sign. For these operations we have:
1) a + 1 = A"; 1) AÅ =a"\
2) a + b" = (a + b)" 2) AÅ b" = (aÅ b)".
Let's prove that
("a, bÎ N )a + b=aÅ b. (1)
Let the number A chosen at random, and b M b, for which equality (1) is true.
It is easy to verify that 1 О M. Indeed, from the fact that A+ 1 = A"=AÅ 1 it follows that a + 1 =aÅ 1.
Let us now prove that if bÎ M, That b" О М, those. If a + b = aÅ b, That A+ b" = aÅ b". Because a + b - aÅ b, then according to axiom 2 (a + b)" = (aÅ b)", and then a + b" - (a + b)" = (aÅ b)" = aÅ b". Since many M contains 1 and together with each number b also contains a number b¢ then by axiom 4, the set M coincides with N, which means equality (1) b. Since the number A was chosen arbitrarily, then equality (1) is true for any natural A And b, those. operations + and Å on a set N may differ from each other only in designations.
Proof of the existence of addition. Let us show that an algebraic operation with properties 1 and 2 specified in the definition of addition exists.
Let M - the set of those and only those numbers A, for which it is possible to determine a + b so that conditions 1 and 2 are satisfied. Let us show that 1 О M. To do this, for any b let's put
1+b=b¢.(2)
1)1 + 1 = 1¢ - according to rule (2), i.e. equality holds a + 1 = A" at A= 1.
2)1 + b"= (b")¢b= (1 + b)" - according to rule (2), i.e. the equality a + b"= (a + b)" at a = 1.
So 1 belongs to the set M.
Let's pretend that A belongs M. Based on this assumption, we will show that A" contained in M, those. that addition can be defined A" and any number b so that conditions 1 and 2 are satisfied. To do this, we set:
A"+ b =(a + b)".(3)
Since by assumption the number a + b is defined, then by axiom 2, the number is also determined in a unique way (A+ b)". Let's check that conditions 1 and 2 are met:
1)a" + 1 = (a + 1)" = (A")". Thus, A"+ 1 = (a")".
2)a" + b" = (a+ b¢)"= ((a + b)")"= (a" + b)". Thus, a" + b" = = (a" + b)".
So, we showed that the set M contains 1 and together with each number A contains a number A". According to axiom 4, we conclude that the set M there are many natural numbers. Thus, there is a rule that allows for any natural numbers A And b uniquely find such a natural number a + b, that properties 1 and 2 formulated in the definition of addition are satisfied.
Let us show how from the definition of addition and Theorem 3 one can derive the well-known table for adding single-digit numbers.
Let's agree on the following notation: 1" = 2; 2" = 3; 3¢ =4; 4"=5, etc.
We compile a table in the following sequence: first we add one to any single-digit natural number, then the number two, then three, etc.
1 + 1 = 1¢ based on property 1 of the definition of addition. But we agreed to denote 1¢ as 2, therefore 1 + 1 = 2.
Similarly 2+1=2" = 3; 3 + 1=3" = 4, etc.
Let us now consider cases involving the addition of the number 2 to any single-valued natural number.
1+2 = 1 + 1¢ - we used the accepted notation. But 1 + 1¢ = = (1 + 1)" according to property 2 from the definition of addition, 1 + 1 is 2, as stated above. Thus,
1 +2 = 1 + 1" = (1 +1)" = 2" = 3.
Similarly 2 + 2 = 2 + 1" = (2 + 1)" = 3" = 4; 3 + 2 = 3 + 1¢= (3 + 1)" = = 4" = 5, etc.
If we continue this process, we get the entire table of adding single-digit numbers.
The next step in the axiomatic construction of a system of natural numbers is the proof of the properties of addition, and the property of associativity is considered first, then commutativity, etc.
Theorem 4.(" a,b,cО N )(a + b)+ With= A+ (b+ With).
Proof. Let the natural numbers A And b chosen randomly, and With takes on various natural meanings. Let us denote by M the set of all those and only those natural numbers c for which the equality (a+b) +c = a+(b+c) right.
Let us first prove that 1 О M, those. let's make sure the equality is fair (A+ b)+ 1 = A+ (b+ 1) Indeed, by the definition of addition, we have (a + b)+ 1 = (A+ b)"= A+ b"= A+ (b+ 1).
Let us now prove that if c О М, then c" О M, those. from equality (A+ b)+ c = a+ (b + c) equality follows (A+ b)+ With"= A+ (b + c"). (A+ b)+ With"= ((A + b)+ With)". Then, based on the equality (A+ b) + c= a + (b + c) can be written: ((A+ b)+ c)" = (a+ (b+ With))". From where, by the definition of addition, we get: ( a + (b+ c))" = a + (b + c)" = a + (b + c") .
M contains 1, and from the fact that With contained in M, it follows that With" contained in M. Therefore, according to axiom 4, M= N, those. equality ( A + b)+ With= a + (b + c) true for any natural number With, and since the numbers A And b were chosen arbitrarily, then it is true for any natural numbers A And b, Q.E.D.
Theorem 5.(" a, bÎ N) a+ b= b+ A.
Proof. It consists of two parts: first they prove that (" aО N) A+1 = 1+a and then what(" a, bО N ) a + b=b+ A.
1 .Let us prove that (" A ON) a+ 1=1+a. Let M - the set of all those and only those numbers A, for which equality A+ 1 = 1 + A true.
Since 1+1=1 + 1 is a true equality, then 1 belongs to the set M.
Let us now prove that if AÎ M, That A"Î M, i.e. from equality a + 1 = 1 + A equality follows a" + 1 = 1 + A". Really, a" + 1 = (a + 1) + 1 by the first property of addition. Next, the expression (a + 1) + 1 can be converted into the expression (1 + a) + 1, using the equality A+ 1 = 1 + A. Then, based on the associative law, we have: (1 + A)+ 1 = 1 + (A+ 1). And finally, by the definition of addition, we get: 1 +(a + 1) = 1 +a".
Thus, we have shown that the set M contains 1 and together with each number A also contains a number A". Therefore, according to the axiom A, M = I, those. equality A+ 1 = 1 + A true for any natural A.
2 . Let us prove that (" a, bÎ N ) A+ b = b+ A. Let A - an arbitrarily chosen natural number, and b takes on various natural meanings. Let us denote by M the set of all those and only those natural numbers b, for which equality a + b =b+ A true.
Since when b = 1 we get the equality A+ 1 = 1 + A, the truth of which is proven in paragraph 1, then 1 is contained in M.
Let us now prove that if b belongs M, then and b" also belongs M, those. from equality A+ b =b+ A equality follows A+ b"= b"+ A. Indeed, by the definition of addition, we have: A+ b"= (A+ b)". Because A+ b= b+ A, That (A+ b)" =(b+ A)". Hence, by definition of addition: (b+ A)"= b+ A"= b+ (a+ 1). Based on the fact that a + 1 = 1 + A, we get: b+ (a + 1) = b+ (1 + A). Using the associative property and the definition of addition, we perform the transformations: b + (1 + a) = (b+1) + a = b" + a.
So, we have proven that 1 is contained in the set M and along with each number b a bunch of M also contains a number b¢, immediately following b¢. By axiom 4 we get that M= AND, those. equality a+ b= b+ A true for any natural number b, as well as for any natural A, because his choice was arbitrary.
Theorem 6.("a,bÎ N) a + b¹ b.
Proof. Let A - a natural number chosen at random, and b takes on various natural meanings. Let us denote by M the set of those and only those natural numbers b, for which Theorem 6 is true.
Let us prove that 1 О M. Indeed, since A+ 1 = A"(by definition of addition), and 1 does not follow any number (axiom 1), then A+ 1 ¹ 1.
Let us now prove that if bÎ M, That b"Î M, those. from what a +bÎ b it follows that a + b"¹ b". Indeed, by the definition of addition, a + b" = (a + b)", but because a +bÎ b, That (a + b)"¹ b" and, therefore, a +b¢=b¢.
According to the axiom there are 4 sets M And N coincide, therefore, for any natural numbers a +bÎ b, Q.E.D.
The approach to addition, considered in the axiomatic construction of the system of natural numbers, is the basis of initial mathematics education. Obtaining numbers by adding 1 is closely related to the principle of constructing the natural series, and the second property of addition is used in calculations, for example, in the following cases: 6 + 3 = (6+ 2)+ 1=8 + 1= 9.
All proven properties are studied in the initial mathematics course and are used to transform expressions.
Exercises
1. Is it true that each natural number is obtained from the previous one by adding one?
2. Using the definition of addition, find the meaning of the expressions:
a) 2 + 3; b) 3 + 3; c) 4 + 3.
3. What transformations of expressions can be performed using the associativity property of addition?
4. Transform an expression using the associative property of addition:
a) (12 + 3)+17; b) 24 + (6 + 19); c) 27+13+18.
5. Prove that (" a, bÎ N) a + b¹ A.
6. Find out how mathematics is formulated in various elementary school textbooks:
a) the commutative property of addition;
b) associative property of addition.
7 .One of the textbooks for elementary school examines the rule for adding a number to a sum using a specific example (4 + 3) + 2 and suggests the following ways to find the result:
a) (4 + 3) + 2 = 7 + 2 = 9;
b) (4 + 3) + 2 = (4 + 2) + 3 = 6 + 3 = 9;
c) (4 + 3) + 2 = 4 + (2 + 3) = 4 + 5 = 9.
Justify the transformations performed. Is it possible to say that the rule for adding a number to a sum is a consequence of the associative property of addition?
8 .It is known that a + b= 17. What is equal to:
A) a + (b + 3); b) (A+ 6) + b; c) (13+ b)+a?
9 .Describe possible ways to calculate the value of an expression of the form a + b + c. Give justification for these methods and illustrate them with specific examples.
Multiplication
According to the rules for constructing an axiomatic theory, the multiplication of natural numbers can be determined using the “directly follow” relation and the concepts introduced earlier.
Let us preface the definition of multiplication with the following considerations. If any natural number A multiply by 1, you get A, those. there is equality a× 1 = A and we get the rule for multiplying any natural number by 1. But how to multiply a number A to a natural number b, different from 1? Let's use the following fact: if we know that 7×5 = 35, then to find the product 7×6 it is enough to add 7 to 35, since 7×6=7×(5 + 1) = 7×5 +7. Thus, the work a×b" can be found if the work is known: a×b" = a×b+ A.
The noted facts form the basis for the definition of multiplication of natural numbers. In addition, it uses the concept of algebraic operation.
Definition. Multiplication of natural numbers is an algebraic operation that has the following properties:
1) ("a Î N) a× 1= a;
2) ("a, Î N) a×b"= а×b+ A.
Number а×b called work numbers A And b, and the numbers themselves A And b-multipliers.
The peculiarity of this definition, as well as the definition of addition of natural numbers, is that it is not known in advance whether there exists an algebraic operation that has the indicated properties, and if it exists, then whether it is unique. In this regard, there is a need to prove this fact.
Theorem 7. Multiplication of natural numbers exists, and it is unique.
The proof of this theorem is similar to the proof of Theorem 3.
Using the definition of multiplication, Theorem 7 and the addition table, You can derive a multiplication table for single-digit numbers. We do this in the following sequence: first we consider multiplication by 1, then by 2, etc.
It is easy to see that multiplication by 1 is performed by property 1 in the definition of multiplication: 1×1 = 1; 2×1=2; 3×1=3, etc.
Let us now consider the cases of multiplication by 2: 1×2 = 1×1"= 1×1 + 1 = 1 + 1=2 - the transition from the product 1×2 to the product 1×1¢ is carried out according to the previously accepted notation; the transition from expression 1 ×1 to the expression 1×1+1 - based on the second property of multiplication, the product 1×1 is replaced by the number 1 according to the result already obtained in the table, and finally the value of the expression 1+1 is found according to the addition table. Similarly:
2×2 = 2×1" = 2×1 +2 = 2 + 2 = 4;
3×2 = 3×1¢ = 3×1 + 3 = 3 + 3 = 6.
If we continue this process, we get the entire multiplication table for single-digit numbers.
As is known, multiplication of natural numbers is commutative, associative and distributive with respect to addition. When building a theory axiomatically, it is convenient to prove these properties, starting with distributivity.
But due to the fact that the property of commutativity will be proved later, it is necessary to consider distributivity on the right and on the left with respect to addition.
Theorem 8. ("a,b,cÎ N) (A+ b)×c =a×c+ b×с.
Proof. Let natural numbers a and b chosen randomly, and With takes on various natural meanings. Let us denote by M the set of all those and only those natural numbers c for which the equality (a + b)×c = a×c+ b×с.
Let us prove that 1 О M, those. that equality ( a + b)× 1 = A×1+ b× 1 true. According to property 1 from the definition of multiplication we have: (a + b)× 1=a+b=a× 1+ b×1.
Let us now prove that if WithÎ M, That With"Î M, those. which from the equality ( a + b)c = a×c+ b×с equality follows (A+ b)×c" = a×c"+ b×с". By definition of multiplication, we have: ( a + b)×c"= (a + b)×s+ (a + b). Because (a + b)×c=a×c + b×c, That ( a + b)×c+ (a+b)= (a×c + b×c) + (a+ b). Using the associative and commutative property of addition, we perform the transformations: ( a× With+ b×с)+ (A+ b) =(a× With + b×с+ A)+ b =(a×c + a + b×с)+ b= = ((a×c+ a) + b×с)+ b = (a×c+ a) + (b×с+ b). And finally, by the definition of multiplication, we get: (a×c+ a) + (b×с+ b) =a×c"+ b×с".
So, we have shown that the set M contains 1, and since it contains c, it follows that With" contained in M. By axiom 4 we get that M= N. This means that the equality ( a + b)×c = a×c + b×c true for any natural numbers With, as well as for any natural a And b, since they were chosen randomly.
Theorem 9. (" a, b, cÎ N) a×(b + c) =a×b + a×c.
This is the property of left distributivity with respect to addition. It is proved in a similar way to how it was done for right distributivity.
Theorem 10.(" a,b,cÎ N)(a×b)×c=a×(b×c).
This is the associative property of multiplication. His proof is based on the definition of multiplication and Theorems 4-9.
Theorem 11. ("a,b,Î N) a×b.
The proof of this theorem is similar in form to the proof of the commutative property of addition.
The approach to multiplication, considered in axiomatic theory, is the basis for teaching multiplication in elementary school. Multiplication by 1 is generally defined, and the second property of multiplication is used in single-digit multiplication tables and calculations.
In the initial course, we study all the properties of multiplication that we have considered: commutativity, associativity, and distributivity.
Exercises
1 . Using the definition of multiplication, find the meanings of the expressions:
a) 3×3; 6) 3x4; c) 4×3.
2. Write down the left distributive property of multiplication with respect to addition and prove it. What expression transformations are possible based on it? Why did it become necessary to consider the left and right distributivity of multiplication relative to addition?
3. Prove the associative property of multiplication of natural numbers. What expression transformations are possible based on it? Is this property taught in elementary school?
4. Prove the commutative property of multiplication. Give examples of its use in an elementary mathematics course.
5. What properties of multiplication can be used when finding the value of an expression:
a) 5×(10 + 4); 6)125×15×6; c) (8×379)×125?
6. It is known that 37 - 3 = 111. Using this equality, calculate:
a) 37×18; b)185×12.
Justify all transformations performed.
7 . Determine the value of an expression without performing written calculations. Justify your answer:
a) 8962×8 + 8962×2; b) 63402×3 + 63402×97; c) 849+ 849×9.
8 . What properties of multiplication will primary school students use when completing the following tasks:
Is it possible, without calculating, to say which expressions will have the same values:
a) 3×7 + 3×5; b) 7×(5 + 3); c) (7 + 5)×3?
Are the equalities true:
a) 18×5×2 = 18× (5×2); c) 5×6 + 5×7 = (6 + 7)×5;
b) (3×10)×17 = 3×10×17; d) 8×(7 + 9) = 8×7 + 9×8?
Is it possible to compare the values of the expressions without performing calculations:
a) 70×32+ 9×32... 79×30 + 79×2;
b) 87×70 + 87×8 ... 80×78 +7×78?