Examples of induction. Method of mathematical induction: examples of solutions
True knowledge at all times was based onthe establishment of regularity and the proof of its truthfulness in certain circumstances. For such a long period of existence of logical reasoning, the rules were formulated, and Aristotle even compiled a list of "correct reasoning". Historically, it is common to divide all conclusions into two types - from concrete to plural (induction) and vice versa (deduction). It should be noted that the types of evidence from the private to the general and from the general to the particular exist only in the relationship and can not be used interchangeably.
Induction in mathematics
The term "induction" has Latinroots and is literally translated as "guidance". With close study, we can distinguish the structure of the word, namely, Latin prefix - in- (denotes directed action inside or being inside) and -duction - introduction. It should be noted that there are two types - complete and incomplete induction. A complete form is characterized by conclusions drawn from the study of all objects of a certain class.
Incomplete - the conclusions applied to all subjects of the class, but made based on the study of only a few units.
Complete mathematical induction - inference,based on the general conclusion about the whole class of any objects, functionally connected by the relations of a natural number of numbers on the basis of knowledge of this functional connection. The process of proof goes through three stages:
- the first proves the correctness of the position of mathematical induction. Example: f = 1, this is the basis of induction;
- The next stage is based on the assumption of the validity of the position for all natural numbers. That is, f = h, this is the induction hypothesis;
- the third stage proves justiceposition for the number f = h + 1, on the basis of the correctness of the previous clause, is an induction step, or a step of mathematical induction. An example is the so-called "domino principle": if the first bone falls in a row (basis), then all the bones in the row (transition) will fall.
And for fun, and seriously
For simplicity of perception, examples of solving by mathematical induction method are exposed in the form of joke problems. This is the task of "polite turn":
- Rules of conduct prohibit a man from occupyingturn in front of a woman (in this situation, she is allowed to go ahead). Proceeding from this statement, if the last in line is a man, then all the rest are men.
A striking example of the method of mathematical induction is the problem of "dimensionless flight":
- It is required to prove that the minibus is placedany number of people. It is true that one person can be accommodated inside the transport without any difficulties (basis). But no matter how busy the minibus is, one passenger will always fit into it (induction step).
Familiar Circles
Examples of the solution by mathematical induction of problems and equations are quite often encountered. As an illustration of this approach, we can consider the following problem.
Condition: on the plane there are h circles. It is required to prove that for any arrangement of figures the card they form can be correctly colored with two colors.
Decision: for h = 1 the truth of the assertion is obvious, therefore the proof will be based on the number of circles h + 1.
We assume that the statement is reliable forany map, and on the plane given h + 1 circles. Removing one of the circles from the total, you can get correctly colored with two colors (black and white) map.
When restoring a deleted circle, thethe color of each region is the opposite (in this case inside the circle). A map is obtained, correctly colored with two colors, which was to be proved.
Examples with natural numbers
Below is clearly shown the application of the method of mathematical induction.
Examples of solutions:
Prove that for any h the following equality holds:
12+22+32+ ... + h2= h (h + 1) (2h + 1) / 6.
Decision:
1. Let h = 1, then:
R1= 12= 1 (1 + 1) (2 + 1) / 6 = 1
It follows that for h = 1 the assertion is correct.
2. Assuming that h = d, we obtain the equation:
R1= d2= d (d + 1) (2d + 1) / 6 = 1
3. Assuming that h = d + 1, we obtain:
Rd + 1= (d + 1) (d + 2) (2d + 3) / 6
Rd + 1= 12+22+32+ ... + d2+ (d + 1)2= d (d + 1) (2d + 1) / 6 + (d + 1)2= (d (d + 1) (2d + 1) + 6 (d + 1)2) / 6 = (d + 1) (d (2d + 1) + 6 (k + 1)) / 6 =
(d + 1) (2d2+ 7d + 6) / 6 = (d + 1) (2 (d + 3/2) (d + 2)) / 6 = (d + 1) (d + 2) (2d + 3) / 6.
Thus, the validity of the equality for h = d + 1 is proved, therefore the assertion is true for any natural number, which is shown in the example of the solution by mathematical induction.
A task
Condition: it is required to prove that for any value of h the expression 7h-1 is divisible by 6 without remainder.
Decision:
1. Assume, h = 1, in this case:
R1= 71-1 = 6 (i.e., divisible by 6 without remainder)
Therefore, for h = 1 the statement is valid;
2. Suppose that h = d and γd-1 is divisible by 6 without residue;
3. A proof of the validity of the assertion for h = d + 1 is the formula:
Rd+1= 7d+1-1 = 7 ∙ 7d-7 + 6 = 7 (7d-1) +6
In this case the first term is divided by 6 according to the assumption of the first point, and the second term is equal to 6. The assertion that 7h-1 is divisible by 6 without remainder for any natural h - it is true.
Mistake of judgments
Often in the evidence used incorrectreasoning, because of the inaccuracy of the logical constructions used. Basically, this happens when the structure and logic of the proof are violated. An example of an incorrect reasoning can serve as an illustration.
A task
Condition: you need proof that any heap of stones is not a bunch.
Decision:
1. Assume, h = 1, in this case in stone 1 the stone and the statement is true (basis);
2. Suppose that for h = d it is true that a pile of stones is not a pile (assumption);
3. Let h = d + 1, which implies that when adding one more stone, the set will not be a heap. The conclusion is that the assumption is valid for all natural h.
The error lies in the fact that there is no definition of how many stones form a pile. Such an omission is called a hasty generalization in the method of mathematical induction. An example of this clearly shows.
Induction and the laws of logic
Historically, examples of induction and deduction always "go hand in hand." Such scientific disciplines as logic, philosophy describe them in the form of opposites.
From the point of view of the law of logic in inductivedefinitions can be viewed based on facts, and the veracity of the premises does not determine the correctness of the resulting statement. Often, conclusions are obtained with a certain degree of probability and plausibility, which, naturally, must be verified and confirmed by additional studies. An example of induction in logic is the statement:
In Estonia - drought, in Latvia - drought, in Lithuania - drought.
Estonia, Latvia and Lithuania are the Baltic states. In all the Baltic States drought.
From the example we can conclude that the new informationor truth cannot be obtained by the method of induction. All that can be expected is some possible truthfulness of the conclusions. Moreover, the truth of the package does not guarantee the same conclusions. However, this fact does not mean that induction stays on the margins of deduction: a great many provisions and scientific laws are substantiated using the method of induction. An example is the same mathematics, biology and other sciences. This is due mostly to the method of full induction, but in some cases partial is applicable.
The respectable age of induction allowed it to penetrate almost all spheres of human activity - this is science, and economics, and everyday conclusions.
Induction in the scientific environment
The induction method requires a scrupulous relationship,since too much depends on the number of studied particulars of the whole: the larger the number studied, the more reliable the result. Proceeding from this feature, the scientific laws obtained by the method of induction are checked for a rather long time at the level of probabilistic assumptions for isolating and studying all possible structural elements, bonds and influences.
In science, induction inference is based onsignificant features, with the exception of random provisions. This fact is important in connection with the specifics of scientific knowledge. This is clearly seen in the examples of induction in science.
There are two types of induction in the scientific world (in connection with the mode of study):
- induction-selection (or selection);
- induction - an exception (elimination).
The first type is distinguished by methodical (scrupulous) sampling of class (subclasses) from different areas of it.
An example of the induction of this type is the following: silver (or silver salts) purifies water. The conclusion is based on long-term observations (a kind of selection of evidence and refutations - selection).
The second kind of induction is built on the conclusionsestablishing causal relationships and excluding circumstances that do not correspond to its properties, namely, universality, adherence to temporal sequence, necessity and uniqueness.
Induction and deduction from the perspective of philosophy
If you look at the historical retrospective,The term "induction" was first mentioned by Socrates. Aristotle described examples of induction in philosophy in a more approximate terminological dictionary, but the question of incomplete induction remains open. After the persecutions of the Aristotle syllogism, the inductive method became recognized as fruitful and the only possible method in natural science. Bacon is considered the father of induction as an independent special method; however, he was not able to separate induction from the deductive method, as contemporaries demanded.
Further development of induction involved J. Mill, who considered induction theory from the perspective of four main methods: agreement, difference, residuals and corresponding changes. Not surprisingly, to date, these methods, when they are examined in detail, are deductive.
Awareness of the failure of the theories of Bacon and Millled scientists to research the probabilistic basis of induction. However, this was not without extremes: attempts were made to reduce induction to probability theory with all the ensuing consequences.
The vote of confidence induction gets at a practicalapplication in certain subject areas and due to the metric accuracy of the inductive base. An example of induction and deduction in philosophy can be considered the law of universal aggression. On the date of the discovery of the law, Newton was able to verify it with an accuracy of 4 percent. And when checking after more than two hundred years, correctness was confirmed with an accuracy of up to 0.0001 percent, although the check was carried out by the same inductive generalizations.
Modern philosophy pays more attentiondeduction, which is dictated by the logical desire to derive from the already known new knowledge (or truth), without resorting to experience, intuition, but in terms of “pure” reasoning. When addressing the true premises in the deductive method in all cases, the output is a true statement.
This very important characteristic should notovershadow the value of the inductive method. Since induction, based on the achievements of experience, becomes the means of its processing (including generalization and systematization).
The use of induction in economics
Induction and deduction have long been used as methods of researching the economy and forecasting its development.
The spectrum of using the induction method is sufficientwide: a study of the implementation of forecast indicators (profits, depreciation, etc.) and the overall assessment of the state of the enterprise; the formation of an effective policy of promoting an enterprise based on facts and their interrelations.
The same method of induction is applied in the “Shewhart maps”, where under the assumption of the separation of processes into controllable and uncontrollable, it is argued that the framework of the controlled process is not very mobile.
It should be noted that scientific lawssubstantiated and supported by the method of induction, and since economics is a science, often using mathematical analysis, risk theory and statistical data, it is not surprising that the presence of induction in the list of basic methods.
An example of induction and deduction in economics canserve the following situation. The increase in the price of food (from the consumer basket) and essential goods pushes the consumer to think about the emerging cost of living in the state (induction). At the same time, it is possible to derive price growth indices for certain goods or categories of goods (deduction) from the fact of high prices with the help of mathematical methods.
Most often refers to the method of inductionmanagement staff, managers, economists. In order to be able to predict with sufficient truthfulness the development of the enterprise, the market behavior, the effects of competition, an inductive-deductive approach to the analysis and processing of information is necessary.
A good example of induction in economics, referring to erroneous judgments:
- the company's profit declined by 30%;
a competing company has expanded its product line;
nothing else has changed; - production policy of a competing company was the reason for reducing profits by 30%;
- therefore, the same production policy is required.
An example is a colorful illustration of how the inept use of the induction method contributes to the ruin of an enterprise.
Deduction and induction in psychology
Since there is a method, then, logically,properly organized thinking takes place (to use the method). Psychology as a science that studies mental processes, their formation, development, interconnections, interactions, pays attention to “deductive” thinking, as one of the forms of deduction and induction. Unfortunately, on the pages of psychology on the Internet there is practically no justification for the integrity of the deductive-inductive method. Although professional psychologists are more often faced with manifestations of induction, or rather, erroneous conclusions.
An example of induction in psychology, as illustrationserroneous judgments, the following statement may serve: my mother is deceiving, therefore, all women are deceivers. Even more can be found "erroneous" examples of induction from life:
- the student is not capable of anything, if he got a deuce in mathematics;
- he is a fool;
- he is smart;
- I can do anything;
- and many other evaluative judgments derived from absolutely random and, at times, insignificant messages.
It should be noted: when the fallacy of a person's judgments reaches the point of absurdity, the front of work for the psychotherapist appears. One example of induction at a specialist appointment:
"The patient is absolutely sure that the color redcarries for him only danger in any manifestations. As a result, a person has excluded a given color scheme from his life - as far as possible. At home there are plenty of opportunities for comfortable living. You can opt out of all the red objects or replace them with analogues made in a different color scheme. But in public places, at work, in the store - is impossible. Getting into a situation of stress, the patient every time experiences a “tide” of completely different emotional states, which can be dangerous for others. ”
This example of induction, and unconsciously,called "fixed ideas." If this happens to a mentally healthy person, you can talk about the lack of organization of mental activity. Elemental development of deductive thinking can become a way of getting rid of obsessive states. In other cases, psychiatrists work with such patients.
The examples of induction indicate that "ignorance of the law does not exempt from the consequences (erroneous judgments)."
Psychologists, working on the topic of deductive thinking, made a list of recommendations designed to help people master this method.
The first item is the solution of problems. As you can see, the form of induction, which is used in mathematics, can be considered "classical", and the use of this method contributes to the "discipline" of the mind.
The next condition for the development of deductive thinkingis the expansion of horizons (who clearly thinks, clearly states). This recommendation directs the "afflicted" to the science and information shelter (libraries, websites, educational initiatives, travel, etc.).
Accuracy is the next recommendation. After all, from the examples of using induction methods it is clearly seen that it is she who is in many respects a guarantee of the truth of the statements.
They did not bypass the flexibility of the mind, implying the possibility of using different ways and approaches in solving the problem, as well as taking into account the variability of events.
And, of course, observation, which is the main source of accumulation of empirical experience.
We should also mention the so-called"Psychological induction." This term, although infrequently, can be found on the Internet. All sources do not give at least a brief formulation of the definition of this term, but refer to "life examples", while giving out a suggestion, some forms of mental illness, or extreme states of the human psyche as an induction. From all of the above, it is clear that an attempt to deduce a “new term”, relying on false (often not corresponding to reality) assumptions, dooms the experimenter to receive erroneous (or hasty) statements.
It should be noted that the reference to the experiments1960 (without specifying the venue, the names of the experimenters, the sample of subjects and, most importantly, the purpose of the experiment) looks, to put it mildly, unconvincing, and the statement that the brain perceives information, bypassing all organs of perception (the phrase "is affected" in this case would fit more organically), makes you think about the credulity and uncriticality of the author of the statement.
Instead of concluding
The Queen of Sciences - Mathematics, not for nothing that uses allpossible reserves of the method of induction and deduction. The considered examples allow to conclude that the superficial and inept (mindless, as they say) the use of even the most accurate and reliable methods always leads to erroneous results.
In the mass consciousness, the deduction method is associated with the famous Sherlock Holmes, who in his logical constructions often use examples of induction, using deduction in the right situations.
The article examined examples of the application of these methods in various sciences and spheres of human activity.
