Research work "logical problems". Scientific work: Mathematical logic and common sense logic Relevance of the chosen topic
XI REGIONAL SCIENTIFIC AND PRACTICAL CONFERENCE “KOLMOGOROV READINGS”
Section "Mathematics"
Subject
"Solving Logical Problems"
Municipal budgetary general education
school No. 2 st. Arkhonskaya,
7th grade.
Scientific director
mathematics teacher MBOU secondary school No. 2 st. Arkhonskaya
Trimasova N.I.
"Solving Logical Problems"
7th grade
secondary educational institution
school No. 2, st. Arkhonskaya.
annotation
This work discusses different ways to solve logical problems and a variety of techniques. Each of them has its own area of application. In addition, in the work you can get acquainted with the basic concepts of the direction of “mathematics without formulas” - mathematical logic, and learn about the creators of this science. You can also see the results of the diagnostic “solving logical problems among middle-level students.”
Content
1. Introduction_____________________________________________________ 4
2. The founders of the science of “logic”___________________________ 6
3.How to learn to solve logical problems?______________________ _8
4. Types and methods of solving logical problems______________________ 9
4.1 Problems of the “Who’s Who?” type 9
a) Graph method___________________________________________ 9
b) Tabular method__________________________________________ 11
4.2 Tactical tasks_______________________________________________ 13
a) method of reasoning_______________________________________________ 13
4.3 Problems of finding the intersection or union of sets_________________________________________________ 14
a) Euler circles_____________________________________________ 14
Letter puzzles and star problems__________________ 16
4.5 Truth problems_____________________________________________ 17
4.6 “Hat” type problems_____________________________________________ 18
5. Practical part__________________________________________________________ 19
5.1 Study of the level of logical thinking of middle-level students__________________________________________________________ 19
6. Conclusion_________________________________________________________ 23
7. Literature_________________________________________________________ 24
"Solving Logical Problems"
Krutogolova Diana Alexandrovna
7th grade
Municipal budgetary general education
secondary educational institution
school No. 2, st. Arkhonskaya.
1. Introduction
The development of creative activity, initiative, curiosity, and ingenuity is facilitated by solving non-standard problems.Despite the fact that the school mathematics course contains a large number of interesting problems, many useful problems are not covered. These tasks include logical tasks.
Solving logic problems is very exciting. There seems to be no mathematics in them - no numbers, no functions, no triangles, no vectors, but there are only liars and wise men, truth and lies. At the same time, the spirit of mathematics is felt most clearly in them - half of the solution to any mathematical problem (and sometimes much more than half) is to properly understand the condition, to unravel all the connections between the participating objects.
A mathematical problem invariably helps to develop correct mathematical concepts, to better understand various aspects of the relationships in the surrounding life, and makes it possible to apply the theoretical principles being studied. At the same time, problem solving contributes to the development of logical thinking.
While preparing this work, I settarget - develop your ability to reason and draw correct conclusions. Only solving a difficult, non-standard problem brings the joy of victory. When solving logical problems, you have the opportunity to think about an unusual condition and reason. This arouses and maintains my interest in mathematics. A logical decision is the best way to unleash your creativity.
Relevance. Nowadays, very often a person’s success depends on his ability to think clearly, reason logically and clearly express his thoughts.
Tasks: 1) familiarization with the concepts of “logic” and “mathematical logic”; 2) study of basic methods for solving logical problems; 3) conducting diagnostics to identify the level of logical thinking of students in grades 5-8.
Research methods: collection, study, generalization of experimental and theoretical material
2. The founders of the science of “logic”
Logic is one of the most ancient sciences. It is currently not possible to establish exactly who, when and where first turned to those aspects of thinking that constitute the subject of logic. Some of the origins of logical teaching can be found in India, at the end of the 2nd millennium BC. e. However, if we talk about the emergence of logic as a science, that is, about a more or less systematized body of knowledge, then it would be fair to consider the great civilization of Ancient Greece as the birthplace of logic. It was here in the V-IV centuries BC. e. During the period of rapid development of democracy and the associated unprecedented revival of socio-political life, the foundations of this science were laid by the works of Democritus, Socrates and Plato.
The founder of logic as a science is the ancient Greek philosopher and scientist Aristotle (384-322 BC). He first developed the theory of deduction, that is, the theory of logical inference. It was he who drew attention to the fact that in reasoning we deduce others from some statements, based not on the specific content of the statements, but on a certain relationship between their forms and structures.
Even then, schools were created in Ancient Greece in which people learned to debate. The students of these schools learned the art of searching for the truth and convincing other people that they were right. They learned to select the necessary ones from a variety of facts, build chains of reasoning that connect individual facts with each other, and draw the right conclusions.
Already from these times, it was generally accepted that logic is a science about thinking, and not about objects of objective truth.
The ancient Greek mathematician Euclid (330-275 BC) was the first to attempt to organize the extensive information on geometry that had accumulated by that time. He laid the foundation for the understanding of geometry as an axiomatic theory, and of all mathematics as a set of axiomatic theories.
Over the course of many centuries, various philosophers and entire philosophical schools supplemented, improved and changed Aristotle's logic. This was the first, pre-mathematical, stage in the development of formal logic. The second stage is associated with the use of mathematical methods in logic, which was started by the German philosopher and mathematician G. W. Leibniz (1646-1716). He tried to build a universal language with the help of which disputes between people would be resolved, and then completely “replace all ideas with calculations.”
An important period in the formation of mathematical logic begins with the work of the English mathematician and logician George Boole (1815-1864) “Mathematical Analysis of Logic” (1847) and “Investigations into the Laws of Thought” (1854). He applied to logic the methods of contemporary algebra - the language of symbols and formulas, the composition and solution of equations. He created a kind of algebra - the algebra of logic. During this period, it took shape as propositional algebra and was significantly developed in the works of the Scottish logician A. de Morgan (1806-1871), the English one - W. Jevons (1835-1882), the American one - C. Pierce and others. The creation of the algebra of logic was the final link in the development of formal logic.
A significant impetus to a new period in the development of mathematical logic was given by the creation in the first half of the 19th century by the great Russian mathematician N. I. Lobachevsky (1792-1856) and independently by the Hungarian mathematician J. Bolyai (1802-1860) of non-Euclidean geometry. In addition, the creation of the analysis of infinitesimals led to the need to substantiate the concept of number as a fundamental concept of all mathematics. The paradoxes discovered at the end of the 19th century in set theory completed the picture: they clearly showed that the difficulties of substantiating mathematics were difficulties of a logical and methodological nature. Thus, mathematical logic was faced with problems that did not arise before Aristotle’s logic. In the development of mathematical logic, three directions in the substantiation of mathematics were formed, in which the creators tried in different ways to overcome the difficulties that arose.
3. How to learn to solve logical problems?
Many people only think what they think.
They find the thought process unpleasant:
this requires skill and a certain amount of effort,
Why bother when you can do it without it.
Ogden Nash
Logical ornon-numeric problems constitute a broad class of non-standard problems. This includes, first of all, word problems in which it is necessary to recognize objects or arrange them in a certain order according to existing properties. In this case, some of the statements of the problem conditions may have different truth values (be true or false).
Text logic problems can be divided into the following types:
all statements are true;
not all statements are true;
problems about truth-tellers and liars.
It is advisable to practice solving each type of problem gradually, step by step.
So, we will learn how logic problems can be solved in different ways. It turns out there are several such techniques, they are varied and each of them has its own area of application. After getting acquainted in detail, we will figure out in what cases it is more convenient to use one or another method.
4. Types and methods of solving logical problems
4.1 Problems of the “Who is who?” type
Problems like “Who is who?” very diverse in complexity, content and ability to solve. They are certainly of interest.
a) Graph method
One way is to solve using graphs. A graph is several points, some of which are connected to each other by segments or arrows (in this case, the graph is called oriented). Let us need to establish a correspondence between two types of objects (sets). Dots denote elements of sets, and the correspondence between them - segments. The dashed line will merge two elements that do not correspond to each other.
Problem 1 . Three friends Belova, Krasnova and Chernova met. One of them was wearing a black dress, the other was wearing a red dress, and the third was wearing a white dress. A girl in a white dress says to Chernova: “We need to change dresses, otherwise the color of our dresses does not match our surnames.” Who was wearing what dress?
Solution. Solving the problem is simple if you consider that:
Each element of one set necessarily corresponds to an element of another set, but only one
If an element of each set is connected to all elements (except one) of another set by dashed segments, then it is connected to the latter by a solid segment.
Instead of solid line segments, you can use colored ones, in which case the solution is more colorful,
Let's denote the girls' surnames in the picture with the letters B, Ch, K, and connect the letter B and the white dress with a dotted line, which will mean: “Belova is not in a white dress.” Next we get three more dotted lines corresponding to the minuses in the table. A white dress can only be worn by Krasnova - we will connect the letter K and the white dress with a solid line, which will mean “Krasnova in a white dress,” etc.
In the same way, you can find correspondence between three sets.
Task 2. Three friends met in a cafe: the sculptor Belov, the violinist Chernov and the artist Ryzhov. “It’s wonderful that one of us has white hair, another has black, and the third has red hair, but none of our hair color matches our surname,” the black-haired man remarked. “You’re right,” said Belov. What color is the artist's hair?
Solution. First, all conditions are plotted on the diagram. The solution comes down to finding three solid triangles with vertices in different sets (Fig. 2.).
Belov Chernov Ryzhov
sculptor violinist artist
white black red
The artist is black-haired
When solving, we can get triangles of three types:
a) all sides are continuous segments (solution to the problem);
b) one side is a solid segment, and the others are dashed;
c) all sides are dashed segments.
Thus, it is impossible to obtain a triangle in which two sides are solid segments and the third is a dashed segment.
Task 3. Who where?
Oak,maple, pine, birch, stump!
Hiding behind them, they lurk
Beaver, hare, squirrel, lynx, deer.
Who where? Try to figure it out."
Where is the lynx, neither hare nor beaver
Neither on the left nor on the right - it’s clear.
ANDnext to the squirrel - that’s cunning -
Don't look for them in vain either.
There is no lynx next to the deer.
And there is no hare on the right and on the left.
And the squirrel on the right is where the deer is!
Now start your search with confidence.
And wants to give you advice
A tall stump covered with moss:
- Who where? Find the right trail
A squirrel and a deer will help.
Solution. Let's find the answer using graphs, denoting each animal with a dot and its placement with arrows. All that remains is to count the arrows (Fig.)
Lynx Hare
Squirrel Hare Beaver Deer Squirrel Lynx
Deer Oak Maple Pine Birch Stump
beaver
b) Tabular method
The second way to solve logical problems - using tables - is also simple and intuitive, but it can only be used when it is necessary to establish a correspondence between two sets. It is more convenient when sets have five or six elements.
Task 4. One day, seven married couples gathered at a family celebration. The men's surnames: Vladimirov, Fedorov, Nazarov, Viktorov, Stepanov, Matveev and Tarasov. The women's names are: Tonya, Lyusya, Lena, Sveta, Masha, Olya and Galya.
Solution. When solving the problem, we know that each man has one last name and one wife.
Rule 1: Each row and each column of the table can contain only one matching sign (for example, “+”).
Rule 2: If in a row (or column) all the “places”, except one, are occupied by an elementary prohibition (a discrepancy sign, for example “-”), then you need to put a “+” sign on the free space; if there is already a “+” sign in a row (or column), then the remaining places should be occupied by a “-” sign.
Having drawn a table, you need to place known prohibitions in it based on the conditions of the problem. Having filled out the table according to the conditions of the problem, we immediately obtain solutions: (Fig. 3).
Tonya
Lucy
Lena
Sveta
Masha
Olya
Galya
Vladimirov
Fedorov
Nazarov
Viktorov
Stepanov
Matveev
Tarasov
4.2 Tactical tasks
Solving tactical and set-theoretic problems involves drawing up a plan of action that leads to the correct answer. The difficulty is that the choice must be made from a very large number of options, i.e. these possibilities are not known, they need to be invented.
a) Problems of moving or correctly placing pieces can be solved in two ways: practical (actions in moving pieces, selecting) and mental (thinking about a move, predicting the result, guessing a solution -method of reasoning ).
In the method of reasoning, the following help when solving: diagrams, drawings, short notes, the ability to select information, the ability to use the enumeration rule.
This method is usually used to solve simple logical problems.
Problem 5 . Lena, Olya, Tanya took part in the 100 m race. Lena ran 2 seconds earlier than Olya, Olya ran 1 second later than Tanya. Who came earlier: Tanya or Lena and by how many seconds?
Solution. Let's make a diagram:
Lena Olya Tanya
Answer. Earlier, Lena arrived at 1st.
Let's consider a simple problem.
Problem 6 . Remembering the autumn cross, Squirrels argue for two hours:
The hare won the race.Athe second was a fox!
- No, says another squirrel,
- You to mejokes
The first one, I remember, was a moose!
- “I,” said the important owl,
- I won’t get involved in someone else’s dispute.
But in each of your words
There is one error.
The squirrels snorted angrily.
It became unpleasant for them.
After weighing everything, you decide
Who was first, who was second.
Solution.
Hare - 1 2
Fox - 2
Moose - 1
If we assume that the correct statement is the hare came 1, then the fox 2 is then not true, i.e. in the second group of statements, both options remain incorrect, but this contradicts the condition. Answer: Elk - 1, Fox - 2, Hare - 3.
4.3 Problems of finding the intersection or union of sets (Eulerian circles)
Another type of problem is one in which it is necessary to find some intersection of sets or their union, observing the conditions of the problem.
Let's solve problem 7:
Of the 52 schoolchildren, 23 collect badges, 35 collect stamps, and 16 collect both badges and stamps. The rest are not interested in collecting. How many schoolchildren are not interested in collecting?
Solution. The conditions of this problem are not so easy to understand. If you add 23 and 35, you get more than 52. This is explained by the fact that we counted some schoolchildren twice here, namely those who collect both badges and stamps.To make the discussion easier, let's use Euler circles
There is a big circle in the picturedenotes the 52 students in question; circle 3 depicts schoolchildren collecting badges, and circle M depicts schoolchildren collecting stamps.
The large circle is divided by circles 3 and M into several areas. The intersection of circles 3 and M corresponds to schoolchildren collecting both badges and stamps (Fig.). The part of circle 3 that does not belong to circle M corresponds to schoolchildren who collect only badges, and the part of circle M that does not belong to circle 3 corresponds to schoolchildren who collect only stamps. The free part of the large circle represents schoolchildren who are not interested in collecting.
We will sequentially fill out our diagram, entering the corresponding number in each area. According to the condition, both badges and stamps are collected by 16 people, so at the intersection of circles 3 and M we will write the number 16 (Fig.).
Since 23 schoolchildren collect badges, and 16 schoolchildren collect both badges and stamps, then 23 - 16 = 7 people collect badges alone. In the same way, only stamps are collected by 35 - 16 = 19 people. Let's write numbers 7 and 19 in the corresponding areas of the diagram.
From the picture it is clear how many people are involved in collecting. To find out thisyou need to add the numbers 7, 9 and 16. We get 42 people. This means that 52 - 42 = 10 schoolchildren remain not interested in collecting. This is the answer to the problem; it can be entered into the free field of the large circle.
Euler's method is indispensable for solving some problems, and also greatly simplifies reasoning.
4.4 Letter puzzles and problems with asterisks
Letter puzzles and examples with asterisks are solved by selecting and considering various options.
Such problems vary in complexity and solution scheme. Let's look at one such example.
Problem 8 Solve a number puzzle
CIS
KSI
ISK
Solution. Amount AND+ C
(in the tens place) ends in C, but I ≠ 0 (see the units place). This means I = 9 and 1 ten in the units place is remembered. Now it is easy to find K in the hundreds place: K = 4. For C there is only one possibility left: C = 5.
4.5 Truth problems
We will call problems in which it is necessary to establish the truth or falsity of statements truth problems.
Problem 9 . Three friends Kolya, Oleg and Petya were playing in the yard, and one of them accidentally broke the window glass with a ball. Kolya said: “It wasn’t me who broke the glass.” Oleg said: “Petya broke the glass.” It was later discovered that one of these statements was true and the other was false. Which boy broke the glass?
Solution. Let's assume that Oleg told the truth, then Kolya also told the truth, and this contradicts the conditions of the problem. Consequently, Oleg told a lie, and Kolya told the truth. From their statements it follows that Oleg broke the glass.
Problem 10 Four students - Vitya, Petya, Yura and Sergei - took four first places at the Mathematical Olympiad. When asked what places they took, the following answers were given:
a) Petya - second, Vitya - third;
b) Sergey - second, Petya - first;
c) Yura - second, Vitya - fourth.
Indicate who took what place if only one part of each answer is correct.
Solution. Suppose that the statement “Peter - II” is true, then both statements of the second person are incorrect, and this contradicts the conditions of the problem.
Suppose that the statement “Sergey - II” is true, then both statements of the first person are incorrect, and this contradicts the conditions of the problem.
Suppose that the statement "Jura - II" is true, then the first statement of the first person is false, and the second is true. And the first statement of the second person is incorrect, but the second is correct.
Answer: first place - Petya, second place - Yura, third place - Vitya, fourth place Sergey.
4.6 “Hats” type problems
The most famous problem is about wise men who need to determine the color of the hat on their head. To solve such a problem, you need to restore the chain of logical reasoning.
Problem 11 . “What color are the berets?”
Three friends, Anya, Shura and Sonya, sat in the amphitheater one after another without birets. Sonya and Shura cannot look back. Shura sees only the head of Sonya sitting below her, and Anya sees the heads of both friends. From a box containing 2 white and 3 black berets (all three friends know about this), they took three out and put them on their heads, not to mention what color the beret was; two berets remained in the box. When Anya was asked about the color of the beret they put on her, she was unable to answer. Shura heard Anya’s answer and said that she also could not determine the color of her beret. Based on the answers of her friends, can Sonya determine the color of her beret?
Solution. You can reason this way. From Anya’s answers, both girlfriends concluded that they both could not have two white berets on their heads. (Otherwise Anya would have immediately said that she had a black beret on her head). They have either two black ones, or white and black. However, if Sonya had a white beret on her head, then Shura also said that she did not know which beret she had on her head, then, therefore, Sonya had a black beret on her head.
5. Practical part
Study of the level of logical thinking of middle school students.
In the practical part of the research work, I selected logical problems like:Who is who?
The tasks corresponded to the level of knowledge of the 5th and 6th, 7th and 8th grades, respectively. The students solved these problems, and I analyzed the results. Let us consider the results obtained in more detail.
The following tasks were proposed for grades 5 and 6:
Problem 1. Remembering the autumn cross, Squirrels argue for two hours:
The hare won the race.Athe second was a fox!
- No, says another squirrel,
- You to mejokesthrow these away. The hare was second, of course
The first one, I remember, was a moose!
- “I,” said the important owl,
- I won’t get involved in someone else’s dispute.
But in each of your words
There is one error.
The squirrels snorted angrily.
It became unpleasant for them.
After weighing everything, you decide
Who was first, who was second.
Task 2. Three friends of Belova, Krasnova and Chernova met. One of them was wearing a black dress, the other was wearing a red dress, and the third was wearing a white dress. A girl in a white dress says to Chernova: “We need to change dresses, otherwise the color of our dresses does not match our surnames.” Who was wearing what dress?
Among students in grades 5 and 6, there were 25 people with proposed tasks like “Who is who?” 11 people completed it, including 5 girls and 6 boys. The results of solving logical problems by students in grades 5 and 6 are presented in the figure:
The figure shows that 44% successfully solved both “Who is who?” problems. Almost all students coped with the first task; the second task, using graphs or tables, caused difficulties for the children.
To summarize, we can conclude that, in general, 5th and 6th grade students cope with simpler tasks, but if a little more elements are added in reasoning, then not all of them cope with such tasks.
The following tasks were proposed for 7th and 8th grades:
Problem 1. Lena, Olya, Tanya took part in the 100 m race. Lena ran 2 seconds earlier than Olya, Olya ran 1 second later than Tanya. Who came earlier: Tanya or Lena and by how many seconds?
Problem 2. Three friends met in a cafe: the sculptor Belov, the violinist Chernov and the artist Ryzhov. “It’s wonderful that one of us has white hair, another has black, and the third has red hair, but none of our hair color matches our surname,” the black-haired man remarked. “You’re right,” said Belov. What color is the artist's hair?
Problem 3. Once upon a time, seven married couples gathered at a family holiday. The men's surnames: Vladimirov, Fedorov, Nazarov, Viktorov, Stepanov, Matveev and Tarasov. The women's names are: Tonya, Lyusya, Lena, Sveta, Masha, Olya and Galya.At the evening, Vladimirov danced with Lena and Sveta, Nazarov - with Masha and Sveta, Tarasov - with Lena and Olya, Viktorov - with Lena, Stepanov - with Sveta, Matveev - with Olya. Then they started playing cards. First, Viktorov and Vladimirov played with Olya and Galya, then Stepanov and Nazarov replaced the men, and the women continued the game. And finally, Stepanov and Nazarov played one game with Tonya and Lena.
Try to determine who is married to whom if it is known that at the evening not a single man danced with his wife and not a single married couple sat down at the same time at the table during the game.
In the 7th and 8th grades among 33 people with all the problems like “Who is who?” 18 people completed it, including 8 girls and 10 boys.
The results of solving logical problems by students of the 7th and 8th grades are presented in the figure:
The figure shows that 55% of students coped with all the tasks, 91% completed the first task, 67% successfully solved the second task, and the last task turned out to be the most difficult for the children and only 58% completed it.
Analyzing the results obtained, in general we can say that students in the 7th and 8th grades coped better with solving logical problems. Pupils of the 5th and 6th grades showed worse results, perhaps the reason for this is that solving this type of problem requires a good knowledge of mathematics; pupils of the 5th grade do not yet have experience in solving such problems.
I also conducted social. survey among students in grades 5-8. I asked everyone the question: “Which problems are easier to solve: mathematical or logical? 15 people took part in the survey. 10 people answered - mathematical, 3-logical, 2 - they can’t solve anything. The survey results are shown in the figure:
The figure shows that mathematical problems are easier to solve for 67% of respondents, logical problems for 20%, and 13% will not be able to solve any problem.
6.Conclusion
In this work you got acquainted with logical problems. With what logic is. We have brought to your attention various logical tasks that help develop logical and imaginative thinking.
Any normal child has a desire for knowledge, a desire to test himself. Most often, schoolchildren’s abilities remain undiscovered for themselves, they are not confident in their abilities, and are indifferent to mathematics.
For such students, I propose using logical tasks. These tasks can be considered in club and elective classes.
They must be accessible, awaken intelligence, capture their attention, surprise, awaken them to active imagination and independent decisions.
I also believe that logic helps us cope with any difficulties in our lives, and everything we do should be logically comprehended and structured.
We encounter logic and logical problems not only in school in mathematics lessons, but also in other subjects.
7. Literature
Dorofeev G.V. Mathematics 6th grade.-Enlightenment,: 2013.
Matveeva G. Logical problems // Mathematics. - 1999. No. 25. - P. 4-8.
Orlova E. Solution methods logical problems and number problems //
Mathematics. - 1999. No. 26. - P. 27-29.
4. Sharygin I.F. , Shevkin E.A. Tasks for ingenuity.-Moscow,: Education, 1996.-65 p.
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Ministry of Education of the Orenburg Region
State Autonomous Professional Educational Institution
"Orsk Mechanical Engineering College"
Orsk, Orenburg region
Research
mathematics
«
MATH WITHOUT
FORMULAS, EQUATIONS AND
INEQUALITIES
»
Prepared
:
Thorik Ekaterina
,
group student
15LP
Supervisor:
Marchenko O.V.
.,
math teacher
matiki
Mathematics
–
this is a special world in which formulas play a leading role,
symbols and geometric objects. In research
At work we decided
find out what happens if you remove formulas, equations and
inequality?
The relevance of this study is that
from year to year
Lost interest in mathematics. They don’t like mathematics, especially because
-
for formulas.
In this
In our work we want not only to show the beauty of mathematics, but also
overcome emerging ideas about “dryness” in the minds of students,
formal character, isolation of this science from life and practice.
Purpose of the work: to prove that mathematics will remain complete
advanced science, with
this is interesting and multifaceted, if you remove formulas, equations and
inequalities.
Job objectives:
show that mathematician
A
without formulas, equations and
inequalities
is a complete science
; conduct a survey
both
cha
Yu
working; study
informational
e sources; get acquainted with the main solutions
logical problems.
Assuming that the mathematical formulas
-
just a convenient language
to present the ideas and methods of mathematics, then these ideas themselves can be described,
using familiar and visual images from
surrounding life.
The object of our research was methods for solving mathematical
problems without formulas, equations and inequalities.
Our college students were asked to answer the question: what
what will happen to mathematics if formulas, equations and other
equality?
by choosing one answer from the following options:
a) numbers, numbers, letters will remain b) only theory will remain
c) theorems and proofs will remain d) graphs will remain
e) mathematics will become literature g) nothing will remain
The results of this
survey showed that the majority of students are confident without
formulas, equations and inequalities, mathematics will become literature. We decided
refute this opinion. Without formulas, equations and inequalities in mathematics, in
first of all, there will be logical tasks that
e most often constitute
most of the tasks at the Mathematics Olympiad. Variety of logical
the tasks are very large. There are also many ways to solve them. But the greatest
The following have become widespread: the method of reasoning, the method of tables, the method
graphs, circles Hey
Lera, block method
-
schemes
Method of reasoning
the most primitive way. In this manner
the simplest logical problems are solved. His idea is that we
carry out reasoning using sequentially all the conditions of the problem, and
we come to the conclusion that
will be the answer to the problem.
In this manner
usually solve simple logical problems.
The main technique that is used when solving text logic
tasks is
building tables
. Tables not only allow you to visualize
present condition h
problems or her answer, but they help a lot
make correct logical conclusions when solving a problem.
Graph method.
Graph
-
it is a collection of objects with connections between them.
Objects are represented as vertices, or nodes, of a graph (they are denoted
That
glasses), and connections
-
like arcs or ribs. If the connection is unidirectional
indicated on the diagram by lines with arrows, if the connection between objects
double-sided is indicated in the diagram by lines without arrows.
Euler circle method.
Euler diagrams are used in solving
a large group of logical problems. Conventionally, all these tasks can be divided into three
type. In problems of the first type, it is necessary to symbolically express many
gestures,
shaded on Euler diagrams using the sign
ki of intersection operations,
combinations and additions.
In problems of the second type, Euler diagrams
are used to analyze situations related to class definition. Third type
problems for which Euler diagrams are used,
-
tasks for
logical account.
Block method
-
schemes
.
This type of logical problem solving
included in the course
teaching students of general education institutions a course in computer science.
Programming in the language
Pascal
.
In addition to logical problems in mathematics,
ory to solve simple
mathematical problems you have to do absurd things that go beyond
ra
the limitations of our logic, our thinking.
Absurd
–
in mathematics and logic,
means what
-
then the element has no meaning within the given
theories,
systems or
fields, fundamentally incompatible with them, although the element
which is absurd in this system
it may make sense in another way.
In mathematics, sophisms (skill, skill) are classified into a separate group.
-
a complex conclusion, which, nevertheless, upon superficial examination
seems right.
Without formulas in mathematics, a situation may arise where
the other one can
exist in reality, but has no logical explanation. Such situation
called a paradox. The emergence of paradoxes is not something
-
That
irregular, unexpected, accidental in the history of the development of scientific
thinking. Their appearance is signaled
speaks about the need to revise previous
theoretical ideas, putting forward more adequate concepts, principles
and research methods.
The world of a science like mathematics is not limited to just solving
special type of tasks. Besides all the difficulties,
it has something beautiful and interesting,
sometimes even funny. Mathematical humor, as well as the mathematical world,
sophisticated and special.
Thus, without formulas, equations and inequalities, mathematics will remain
a full-fledged science, at the same time interesting and multifaceted.
Bibliographic list.
Agafonova, I. G. Learning to think: Entertaining logical tasks,
tests and exercises for children. Tutorial [Tex] /
I. G. Agafonova
St. Petersburg
IKF MiM
–
express, 1996.
–
Balayan E.N. 1001 olympiad and entertaining problems
and by
mathematics
[Tex]
/ E.N. Balayan.
-
3
-
e ed.
-
Rostov n/d: Phoenix, 2008.
-
Farkov, A.V. Mathematical Olympiads at school. 5
-
11th grade.
[Tex]/
A. V. Farkov.
-
8
-
e ed., rev. and additional
-
M.: Iris
-
press, 2009.
-
http://www.arhimedes.org/
Tournament named after M. V. Lomonosova (Moscow)
http://olympiads.mccme.ru/turlom/
Attached files
This section of our website presents research paper topics on logic in the form of logical problems, sophisms and paradoxes in mathematics, interesting games on logic and logical thinking. The work supervisor should directly guide and assist the student in his research.
The topics presented below for research and design work on logic are suitable for children who love to think logically, solve non-standard problems and examples, explore paradoxes and mathematical problems, and play non-standard logic games.
In the list below, you can select a logic project topic for any grade in a secondary school, from elementary school to high school. To help you correctly design a mathematics project on logic and logical thinking, you can use the developed requirements for the design of work.
The following topics for logic research projects are not final and may be modified due to the requirements set before the project.
Topics of research papers on logic:
Sample topics for research papers on logic for students:
Interesting logic in mathematics.
Algebra logic
Logic and us
Logics. Laws of logic
Logic box. A collection of entertaining logic problems.
Logical tasks with numbers.
Logic problems
Logic problems "Funny arithmetic"
Logical problems in mathematics.
Logical problems for determining the number of geometric shapes.
Logical tasks for the development of thinking
Logical problems in mathematics lessons.
Logic games
Logical paradoxes
Mathematical logic.
Methods for solving logical problems and methods for composing them.
Simulation of logic problems
Educational presentation "Fundamentals of Logic".
Basic types of logical problems and methods for solving them.
In the footsteps of Sherlock Holmes, or Methods for solving logical problems.
Application of graph theory in solving logical problems.
Problems of four colors.
Solving logical problems
Solving logical problems using the graph method.
Solving logical problems in different ways.
Solving logic problems using graphs
Solving logical problems using diagrams and tables.
Solving logical problems.
Syllogisms. Logical paradoxes.
Logic project topics
Sample topics for logic projects for students:
Sophistry
Sophistry around us
Sophisms and paradoxes
Methods for composing and methods for solving logical problems.
Learning to solve logical problems
Algebra of logic and logical foundations of a computer.
Types of tasks for logical thinking.
Two ways to solve logical problems.
Logic and mathematics.
Logic as a science
Logic riddles.
Introduction. 3
1. Mathematical logic (meaningless logic) and “common sense” logic 4
2. Mathematical judgments and inferences. 6
3. Mathematical logic and “Common sense” in the 21st century. eleven
4. Unnatural logic in the foundations of mathematics. 12
Conclusion. 17
References… 18
The expansion of the area of logical interests is associated with general trends in the development of scientific knowledge. Thus, the emergence of mathematical logic in the middle of the 19th century was the result of centuries-old aspirations of mathematicians and logicians to build a universal symbolic language, free from the “shortcomings” of natural language (primarily its polysemy, i.e. polysemy).
The further development of logic is associated with the combined use of classical and mathematical logic in applied fields. Non-classical logics (deontic, relevant, legal logic, decision-making logic, etc.) often deal with the uncertainty and fuzziness of the objects under study, with the nonlinear nature of their development. Thus, when analyzing rather complex problems in artificial intelligence systems, the problem of synergy between different types of reasoning when solving the same problem arises. Prospects for the development of logic in line with convergence with computer science are associated with the creation of a certain hierarchy of possible reasoning models, including reasoning in natural language, plausible reasoning and formalized deductive conclusions. This can be solved using classical, mathematical and non-classical logic. Thus, we are not talking about different “logics”, but about different degrees of formalization of thinking and the “dimension” of logical meanings (two-valued, multi-valued, etc. logic).
Identification of the main directions of modern logic:
1. general or classical logic;
2. symbolic or mathematical logic;
3. non-classical logic.
Mathematical logic is a rather vague concept, due to the fact that there are also infinitely many mathematical logics. Here we will discuss some of them, paying more tribute to tradition than to common sense. Because, quite possibly, this is common sense... Logical?
Mathematical logic teaches you to reason logically no more than any other branch of mathematics. This is due to the fact that the “logicality” of reasoning in logic is determined by logic itself and can be used correctly only in logic itself. In life, when thinking logically, as a rule, we use different logics and different methods of logical reasoning, shamelessly mixing deduction with induction... Moreover, in life we build our reasoning based on contradictory premises, for example, “Don’t put off until tomorrow what can be done today" and "You'll make people laugh in a hurry." It often happens that a logical conclusion we don’t like leads to a revision of the initial premises (axioms).
Perhaps the time has come to say about logic, perhaps the most important thing: classical logic is not concerned with meaning. Neither healthy nor any other! To study common sense, by the way, there is psychiatry. But in psychiatry, logic is rather harmful.
Of course, when we differentiate logic from sense, we mean first of all classical logic and the everyday understanding of common sense. There are no forbidden directions in mathematics, therefore the study of meaning by logic, and vice versa, in various forms is present in a number of modern branches of logical science.
(The last sentence worked out well, although I won’t attempt to define the term “logical science” even approximately). Meaning, or semantics if you will, is dealt with, for example, by model theory. And in general, the term semantics is often replaced by the term interpretation. And if we agree with philosophers that the interpretation (display!) of an object is its comprehension in some given aspect, then the borderline spheres of mathematics, which can be used to attack the meaning in logic, become incomprehensible!
In practical terms, theoretical programming is forced to take an interest in semantics. And in it, in addition to just semantics, there is also operational, and denotational, and procedural, etc. and so on. semantics...
Let us just mention the apotheosis - THE THEORY OF CATEGORIES, which brought semantics to a formal, obscure syntax, where the meaning is already so simple - laid out on shelves that it is completely impossible for a mere mortal to get to the bottom of it... This is for the elite.
So what does logic do? At least in its most classic part? Logic does only what it does. (And she defines this extremely strictly). The main thing in logic is to strictly define it! Set the axiomatics. And then the logical conclusions should be (!) largely automatic...
Reasoning about these conclusions is another matter! But these arguments are already beyond the bounds of logic! Therefore, they require a strict mathematical sense!
It may seem that this is a simple verbal balancing act. NO! As an example of a certain logical (axiomatic) system, let's take the well-known game 15. Let's set (mix) the initial arrangement of square chips. Then the game (logical conclusion!), and specifically the movement of chips to an empty space, can be handled by some mechanical device, and you can patiently watch and rejoice when, as a result of possible movements, a sequence from 1 to 15 is formed in the box. But no one forbids control mechanical device and prompt it, BASED ON COMMON SENSE, with the correct movements of the chips in order to speed up the process. Or maybe even prove, using for logical reasoning, for example, such a branch of mathematics as COMBINATORICS, that with a given initial arrangement of chips it is impossible to obtain the required final combination at all!
There is no more common sense in that part of logic that is called LOGICAL ALGEBRA. Here LOGICAL OPERATIONS are introduced and their properties are defined. As practice has shown, in some cases the laws of this algebra may correspond to the logic of life, but in others they do not. Because of such inconstancy, the laws of logic cannot be considered laws from the point of view of the practice of life. Their knowledge and mechanical use can not only help, but also harm. Especially psychologists and lawyers. The situation is complicated by the fact that, along with the laws of algebra of logic, which sometimes correspond or do not correspond to life reasoning, there are logical laws that some logicians categorically do not recognize. This applies primarily to the so-called laws of the EXCLUSIVE THIRD and CONTRADICTION.
2. Mathematical judgments and inferences
In thinking, concepts do not appear separately; they are connected with each other in a certain way. The form of connection of concepts with each other is a judgment. In each judgment, some connection or some relationship between concepts is established, and this thereby affirms the existence of a connection or relationship between the objects covered by the corresponding concepts. If judgments correctly reflect these objectively existing dependencies between things, then we call such judgments true, otherwise the judgments will be false. So, for example, the proposition “every rhombus is a parallelogram” is a true proposition; the proposition “every parallelogram is a rhombus” is a false proposition.
Thus, a judgment is a form of thinking that reflects the presence or absence of the object itself (the presence or absence of any of its features and connections).
To think means to make judgments. With the help of judgments, thought and concept receive their further development.
Since every concept reflects a certain class of objects, phenomena or relationships between them, any judgment can be considered as the inclusion or non-inclusion (partial or complete) of one concept in the class of another concept. For example, the proposition “every square is a rhombus” indicates that the concept “square” is included in the concept “rhombus”; the proposition “intersecting lines are not parallel” indicates that intersecting lines do not belong to the set of lines called parallel.
A judgment has its own linguistic shell - a sentence, but not every sentence is a judgment.
A characteristic feature of a judgment is the obligatory presence of truth or falsity in the sentence expressing it.
For example, the sentence “triangle ABC is isosceles” expresses some judgment; the sentence “Will ABC be isosceles?” does not express judgment.
Each science essentially represents a certain system of judgments about the objects that are the subject of its study. Each of the judgments is formalized in the form of a certain proposal, expressed in terms and symbols inherent in this science. Mathematics also represents a certain system of judgments expressed in mathematical sentences through mathematical or logical terms or their corresponding symbols. Mathematical terms (or symbols) denote those concepts that make up the content of a mathematical theory, logical terms (or symbols) denote logical operations with the help of which other mathematical propositions are constructed from some mathematical propositions, from some judgments other judgments are formed, the entirety of which constitutes mathematics as a science.
Generally speaking, judgments are formed in thinking in two main ways: directly and indirectly. In the first case, the result of perception is expressed with the help of a judgment, for example, “this figure is a circle.” In the second case, judgment arises as a result of a special mental activity called inference. For example, “the set of given points on a plane is such that their distance from one point is the same; This means that this figure is a circle.”
In the process of this mental activity, a transition is usually made from one or more interconnected judgments to a new judgment, which contains new knowledge about the object of study. This transition is inference, which represents the highest form of thinking.
So, inference is the process of obtaining a new conclusion from one or more given judgments. For example, the diagonal of a parallelogram divides it into two congruent triangles (first proposition).
The sum of the interior angles of a triangle is 2d (second proposition).
The sum of the interior angles of a parallelogram is equal to 4d (new conclusion).
The cognitive value of mathematical inferences is extremely great. They expand the boundaries of our knowledge about objects and phenomena of the real world due to the fact that most mathematical propositions are a conclusion from a relatively small number of basic judgments, which are obtained, as a rule, through direct experience and which reflect our simplest and most general knowledge about its objects.
Inference differs (as a form of thinking) from concepts and judgments in that it is a logical operation on individual thoughts.
Not every combination of judgments among themselves constitutes a conclusion: there must be a certain logical connection between the judgments, reflecting the objective connection that exists in reality.
For example, one cannot draw a conclusion from the propositions “the sum of the interior angles of a triangle is 2d” and “2*2=4”.
It is clear what importance the ability to correctly construct various mathematical sentences or draw conclusions in the process of reasoning has in the system of our mathematical knowledge. Spoken language is poorly suited for expressing certain judgments, much less for identifying the logical structure of reasoning. Therefore, it is natural that there was a need to improve the language used in the reasoning process. Mathematical (or rather, symbolic) language turned out to be the most suitable for this. The special field of science that emerged in the 19th century, mathematical logic, not only completely solved the problem of creating a theory of mathematical proof, but also had a great influence on the development of mathematics as a whole.
Formal logic (which arose in ancient times in the works of Aristotle) is not identified with mathematical logic (which arose in the 19th century in the works of the English mathematician J. Boole). The subject of formal logic is the study of the laws of the relationship of judgments and concepts in inferences and rules of evidence. Mathematical logic differs from formal logic in that, based on the basic laws of formal logic, it explores the patterns of logical processes based on the use of mathematical methods: “The logical connections that exist between judgments, concepts, etc., are expressed in formulas, the interpretation of which is free from ambiguities that could easily arise from verbal expression. Thus, mathematical logic is characterized by formalization of logical operations, more complete abstraction from the specific content of sentences (expressing any judgment).
Let us illustrate this with one example. Consider the following inference: “If all plants are red and all dogs are plants, then all dogs are red.”
Each of the judgments used here and the judgment that we received as a result of restrained inference seems to be patent nonsense. However, from the point of view of mathematical logic, we are dealing here with a true sentence, since in mathematical logic the truth or falsity of a conclusion depends only on the truth or falsity of its constituent premises, and not on their specific content. Therefore, if one of the basic concepts of formal logic is a judgment, then the analogous concept of mathematical logic is the concept of a statement-statement, for which it only makes sense to say whether it is true or false. One should not think that every statement is characterized by a lack of “common sense” in its content. It’s just that the meaningful part of the sentence that makes up this or that statement fades into the background in mathematical logic and is unimportant for the logical construction or analysis of this or that conclusion. (Although, of course, it is essential for understanding the content of what is being discussed when considering this issue.)
It is clear that in mathematics itself meaningful statements are considered. By establishing various connections and relationships between concepts, mathematical judgments affirm or deny any relationships between objects and phenomena of reality.
3. Mathematical logic and “Common sense” in the 21st century.
Logic is not only a purely mathematical, but also a philosophical science. In the 20th century, these two interconnected hypostases of logic turned out to be separated in different directions. On the one hand, logic is understood as the science of the laws of correct thinking, and on the other hand, it is presented as a set of loosely connected artificial languages, which are called formal logical systems.
For many, it is obvious that thinking is a complex process with the help of which everyday, scientific or philosophical problems are solved and brilliant ideas or fatal delusions are born. Language is understood by many simply as a means by which the results of thinking can be transmitted to contemporaries or left to descendants. But, having connected in our consciousness thinking with the concept of “process”, and language with the concept of “means”, we essentially stop noticing the immutable fact that in this case the “means” is not completely subordinated to the “process”, but depending on our purposeful or unconscious choice of certain or verbal cliches has a strong influence on the course and result of the “process” itself. Moreover, there are many cases where such “reverse influence” turns out to be not only an obstacle to correct thinking, but sometimes even its destroyer.
From a philosophical point of view, the task posed within the framework of logical positivism was never completed. In particular, in his later studies, one of the founders of this trend, Ludwig Wittgenstein, came to the conclusion that natural language cannot be reformed in accordance with the program developed by the positivists. Even the language of mathematics as a whole resisted the powerful pressure of “logicalism,” although many terms and structures of the language proposed by the positivists entered some sections of discrete mathematics and significantly supplemented them. The popularity of logical positivism as a philosophical trend in the second half of the 20th century dropped noticeably - many philosophers came to the conclusion that the rejection of many “illogicalities” of natural language, an attempt to squeeze it into the framework of the fundamental principles of logical positivism entails the dehumanization of the process of cognition, and at the same time and the dehumanization of human culture as a whole.
Many reasoning methods used in natural language are often very difficult to map unambiguously into the language of mathematical logic. In some cases, such a mapping leads to a significant distortion of the essence of natural reasoning. And there is reason to believe that these problems are a consequence of the initial methodological position of analytical philosophy and positivism about the illogicality of natural language and the need for its radical reform. The very original methodological setting of positivism also does not stand up to criticism. To accuse spoken language of being illogical is simply absurd. In fact, illogicality does not characterize the language itself, but many users of this language who simply do not know or do not want to use logic and compensate for this flaw with psychological or rhetorical techniques of influencing the public, or in their reasoning they use as logic a system that is called logic only by misunderstanding. At the same time, there are many people whose speech is distinguished by clarity and logic, and these qualities are not determined by knowledge or ignorance of the foundations of mathematical logic.
In the reasoning of those who can be classified as legislators or followers of the formal language of mathematical logic, a kind of “blindness” in relation to elementary logical errors is often revealed. One of the great mathematicians, Henri Poincaré, drew attention to this blindness in the fundamental works of G. Cantor, D. Hilbert, B. Russell, J. Peano and others at the beginning of our century.
One example of such an illogical approach to reasoning is the formulation of the famous Russell paradox, in which two purely heterogeneous concepts “element” and “set” are unreasonably confused. In many modern works on logic and mathematics, in which the influence of Hilbert's program is noticeable, many statements that are clearly absurd from the point of view of natural logic are not explained. The relationship between “element” and “set” is the simplest example of this kind. Many works in this direction claim that a certain set (let's call it A) can be an element of another set (let's call it B).
For example, in a well-known manual on mathematical logic we will find the following phrase: “Sets themselves can be elements of sets, so, for example, the set of all sets of integers has sets as its elements.” Note that this statement is not just a disclaimer. It is contained as a “hidden” axiom in formal set theory, which many experts consider the foundation of modern mathematics, as well as in the formal system that the mathematician K. Gödel built when proving his famous theorem on the incompleteness of formal systems. This theorem refers to a rather narrow class of formal systems (they include formal set theory and formal arithmetic), the logical structure of which clearly does not correspond to the logical structure of natural reasoning and justification.
However, for more than half a century it has been the subject of heated discussion among logicians and philosophers in the context of the general theory of knowledge. With such a broad generalization of this theorem, it turns out that many elementary concepts are fundamentally unknowable. But with a more sober approach, it turns out that Gödel’s theorem only showed the inconsistency of the program of formal justification of mathematics proposed by D. Hilbert and taken up by many mathematicians, logicians and philosophers. The broader methodological aspect of Gödel's theorem can hardly be considered acceptable until the following question is answered: is Hilbert's program for justifying mathematics the only possible one? To understand the ambiguity of the statement “set A is an element of set B,” it is enough to ask a simple question: “What elements are set B formed from in this case?” From the point of view of natural logic, only two mutually exclusive explanations are possible. Explanation one. The elements of the set B are the names of some sets and, in particular, the name or designation of the set A. For example, the set of all even numbers is contained as an element in the set of all names (or designations) of sets distinguished by some characteristics from the set of all integers. To give a clearer example: the set of all giraffes is contained as an element in the set of all known animal species. In a broader context, the set B can also be formed from conceptual definitions of sets or references to sets. Explanation two. The elements of the set B are the elements of some other sets and, in particular, all the elements of the set A. For example, every even number is an element of the set of all integers, or every giraffe is an element of the set of all animals. But then it turns out that in both cases the expression “set A is an element of set B” does not make sense. In the first case, it turns out that the element of the set B is not the set A itself, but its name (or designation, or reference to it). In this case, an equivalence relation is implicitly established between the set and its designation, which is unacceptable neither from the point of view of ordinary common sense, nor from the point of view of mathematical intuition, which is incompatible with excessive formalism. In the second case, it turns out that set A is included in set B, i.e. is a subset of it, but not an element. Here, too, there is an obvious substitution of concepts, since the relation of inclusion of sets and the relation of membership (being an element of a set) in mathematics have fundamentally different meanings. Russell's famous paradox, which undermined logicians' confidence in the concept of a set, is based on this absurdity - the paradox is based on the ambiguous premise that a set can be an element of another set.
Another possible explanation is possible. Let a set A be defined by a simple enumeration of its elements, for example, A = (a, b). The set B, in turn, is specified by enumerating some sets, for example, B = ((a, b), (a, c)). In this case, it seems obvious that the element of B is not the name of the set A, but the set A itself. But even in this case, the elements of the set A are not elements of the set B, and the set A is here considered as an inseparable collection, which can well be replaced by its name . But if we considered all elements of the sets contained in it to be elements of B, then in this case the set B would be equal to the set (a, b, c), and the set A in this case would not be an element of B, but a subset of it. Thus, it turns out that this version of the explanation, depending on our choice, comes down to the previously listed options. And if no choice is offered, then elementary ambiguity results, which often leads to “inexplicable” paradoxes.
It would be possible not to pay special attention to these terminological nuances if not for one circumstance. It turns out that many of the paradoxes and inconsistencies of modern logic and discrete mathematics are a direct consequence or imitation of this ambiguity.
For example, in modern mathematical reasoning, the concept of “self-applicability” is often used, which underlies Russell’s paradox. In the formulation of this paradox, self-applicability implies the existence of sets that are elements of themselves. This statement immediately leads to a paradox. If we consider the set of all “non-self-applicable” sets, it turns out that it is both “self-applicable” and “non-self-applicable.”
Mathematical logic contributed a lot to the rapid development of information technology in the 20th century, but the concept of “judgment”, which appeared in logic back in the days of Aristotle and on which, as the foundation, rests the logical basis of natural language, fell out of its field of vision. Such an omission did not at all contribute to the development of a logical culture in society and even gave rise to the illusion among many that computers are capable of thinking no worse than humans themselves. Many are not even embarrassed by the fact that against the backdrop of general computerization on the eve of the third millennium, logical absurdities within science itself (not to mention politics, lawmaking and pseudoscience) are even more common than at the end of the 19th century. And in order to understand the essence of these absurdities, there is no need to turn to complex mathematical structures with multi-place relations and recursive functions that are used in mathematical logic. It turns out that to understand and analyze these absurdities, it is quite enough to apply a much simpler mathematical structure of judgment, which not only does not contradict the mathematical foundations of modern logic, but in some way complements and expands them.
Bibliography
1. Vasiliev N. A. Imaginary logic. Selected works. - M.: Science. 1989; - pp. 94-123.
2. Kulik B.A. Basic principles of common sense philosophy (cognitive aspect) // Artificial Intelligence News, 1996, No. 3, p. 7-92.
3. Kulik B.A. Logical foundations of common sense / Edited by D.A. Pospelov. - St. Petersburg, Polytechnic, 1997. 131 p.
4. Kulik B.A. The logic of common sense. - Common Sense, 1997, No. 1(5), p. 44 - 48.
5. Styazhkin N.I. Formation of mathematical logic. M.: Nauka, 1967.
6. Soloviev A. Discrete mathematics without formulas. 2001//http://soloviev.nevod.ru/2001/dm/index.html
MINISTRY OF EDUCATION AND SCIENCE OF THE REPUBLIC OF BURYATIA
MUNICIPAL BUDGET EDUCATIONAL INSTITUTION
"MALOKUDARINSKAYA SECONDARY SCHOOL"
RESEARCH
Topic: “Logical tasks
Completed the job:
Igumnov Matvey, 3rd grade student
MBOU "Malokudarinskaya secondary school"
Head: Serebrennikova M.D.
1. INTRODUCTION …………………………………………………………..3-4
2. MAIN PART
What is logic………………………………………………………. …5
Types of logical problems………………………………………………………6
Solving a logical problem…………………………………………………….10
Practical part …………………………………………………….. 10-12
3. CONCLUSION……………………………………………………… 14
4. LIST OF REFERENCES AND INTERNET SOURCES………. 15
5.APPLICATIONS
Introduction
The development of creative activity, initiative, curiosity, and ingenuity is facilitated by solving non-standard and logical problems.
Solving logic problems is very exciting. There seems to be no mathematics in them - there are no numbers, no geometric figures, but there are only liars and wise men, truth and lies. At the same time, the spirit of mathematics is felt most clearly in them - half of the solution to any mathematical problem (and sometimes much more than half) is to properly understand the condition, to untangle all the connections between the objects of the problem.
While preparing this work, I set target- develop your ability to reason and draw correct conclusions. Only solving a difficult, non-standard problem brings the joy of victory. When solving logical problems, you have the opportunity to think about an unusual condition and reason. This arouses and maintains my interest in mathematics. Relevance. Nowadays, very often a person’s success depends on his ability to think clearly, reason logically and clearly express his thoughts.
Purpose of the study: can a logic problem have multiple correct answers?
Tasks: 1) familiarization with the concepts of “logic” and types of logical problems; 2) solving a logical problem, determining the dependence of the change in the answer of the problem on the size of the nuts
Research methods: collection, study of material, comparison, analysis
Hypothesis If we change the size of the nuts, will the answer to the problem change?
Field of study: logical problem.
What is logic?
The following definitions of logic can be found in the scientific literature:
Logic is the science of acceptable methods of reasoning.
Logic is the science of the forms, methods and laws of intellectual cognitive activity, formalized using logical language.
Logic is the science of correct thinking.
Logic is one of the most ancient sciences. Some of the origins of logical teaching can be found in India, at the end of the 2nd millennium BC. The founder of logic as a science is the ancient Greek philosopher and scientist Aristotle. It was he who drew attention to the fact that in reasoning we deduce others from some statements, based not on the specific content of the statements, but on a certain relationship between their forms and structures.
How to learn to solve logical problems? Logical or non-numeric problems constitute a broad class of non-standard problems. This includes, first of all, word problems in which it is necessary to recognize objects or arrange them in a certain order according to existing properties. In this case, some of the statements of the problem conditions may have different truth values (be true or false). So, we will learn how logic problems can be solved in different ways. It turns out there are several such techniques, they are varied and each of them has its own area of application.
Types of logic problems
1"Who is who?"
2 Tactical tasks Solving tactical and set-theoretic problems involves drawing up a plan of action that leads to the correct answer. The difficulty is that the choice must be made from a very large number of options, i.e. these possibilities are not known, they need to be invented.
3 Problems on finding the intersection or union of sets
4 Letter and number puzzles and star problems
Letter puzzles and examples with asterisks are solved by selecting and considering various options.
5 Tasks that require establishing the truth or falsity of statements
6 “Hats” type problems
The most famous problem is about wise men who need to determine the color of the hat on their head. To solve such a problem, you need to restore the chain of logical reasoning.
SOLVING A LOGICAL PROBLEM
There are many types of nuts. Let's find out whether the answer to this problem depends on the size of the nuts?
Let's look at some of them.
Pine nuts are considered the smallest. Moreover, their sizes depend on the type. The nuts of European cedar, Siberian dwarf cedar and Korean cedar differ in size. Among them, the smallest are dwarf pine nuts. Their length is 5 mm.
Conclusion: There are many types of nuts. They have different sizes: in diameter. Therefore, we substitute nuts of different sizes into the problem.
PRACTICAL PART
Practical work.
Job No. 1. Practical work with walnuts.
Tools and materials: ruler, chalk, colored measures, 10 pieces of walnuts.
Preparatory work. We cut out measurements from colored cardboard: 3 measurements from green cardboard, 2 cm long and 2 cm wide, for the first row and 5 measurements from yellow cardboard, 1 cm long and 2 cm wide, for the second row.
Description of work. Mark a point on the table with chalk. We put a nut on it. Place a 2 cm measure and a second nut, a 2 cm measure and a third nut, a 2 cm measure and a fourth nut. With chalk we mark the beginning and end of the length of the first row. The beginning of the second row is clearly marked with chalk under the beginning
first and put a nut, a 1 cm measure and a second nut, a 1 cm measure and a third, a measure and a fourth, a measure and a fifth, a measure and a sixth. We mark the end of the length of the second row with chalk. Compare the lengths of the rows.
Answer: the second row is longer.
2. Practical work with pine nuts. (See job description #1.)
Answer: the second row is longer.
3. Practical work with hazelnuts (hazelnuts).
(See job description #1.)
Answer:
the second row is longer.
4. Practical work with peanuts. (Fig.4)
(See job description #1.)
Answer: :
the second row is longer.
Conclusion: the answer to the problem does not change depending on the size of these nuts.
All nuts more than 5 mm.
BLUEPRINTS
Let's check this in the drawings using scale.
Scale
1. The ratio of the length of lines on a map or drawing to the actual length.
.
CONCLUSION
My hypothesis was confirmed: when the size of the nuts changes, the answer to the problem changes
Conclusion: For nuts up to 5 mm in size, the first row is longer.
When the nut size is 5 mm, the length of the rows is the same.
For nuts larger than 5 mm, the second row is longer.
Practical significance. The solutions proposed in the work are very simple; any student can use them. I showed them to my friends. Many students became interested in this task. Now, when solving logical problems, everyone will think about its answer.
Prospects: I really enjoyed experimenting with nuts, arranging them, looking for the answer. I shared all my findings with friends and classmates. Logical problems interested me: in the future I want to try to create my own problem that is just as interesting, with different answer options.
I tried changing the problem condition. I took meters for the spaces between the nuts. Substituting nuts of different sizes, I got the same answer: the first row is longer. Why is this so? I started measuring everything again: everything was the same. If I increased the intervals by 100 times, then the size of the nuts should also be increased by 100 times. Now I realized that I don’t have such a large nut of 50 cm or more. All nuts are less than 50 cm. According to my conclusion, for the lengths to be equal, the nut must be 50 cm, and if it is more than 50 cm, then the second row will be longer. This means that my conclusion is also suitable for this task.
6.Conclusion
In this work you got acquainted with logical problems. Various options for solving a logical problem were offered to your attention.
Any normal child has a desire for knowledge, a desire to test himself. Most often, schoolchildren’s abilities remain undiscovered for themselves, they are not confident in their abilities, and are indifferent to mathematics.
For such students, I propose using logical tasks.
They must be accessible, awaken intelligence, capture their attention, surprise, awaken them to active imagination and independent decisions.
I also believe that logic helps us cope with any difficulties in our lives, and everything we do should be logically comprehended and structured.
Literature
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2. Encyclopedia for children. Biology. Volume 2. “Avanta+”, M. Aksenov, S. Ismailova,
M.: “Avanta+”, 1995
3. I explore the world: Det.Entsik.: Plants / Comp. L.A. Bagrova; Khud.A.V.Kardashuk, O.M.Voitenko;
Under general ed. O.G. Hinn. – M.: AST Publishing House LLC, 2000. – 512 p.
4. Encyclopedia of living nature. - M.: AST-PRESS, 2000. - 328 p.
5. Rick Morris. Secrets of living nature (translation from English by A.M. Golov), M.: “Rosman”, 1996.
6. David Burney. Large illustrated encyclopedia of living nature (translation from English) M.: “Swallowtail”, 2006
