Proof in mathematics

One of the oldest fragments Elements of Euclides, a textbook used for thousands of years to teach demo techniques of writing. The diagram accompanies Book II, Proposition 5.1.

In mathematics, a proof or proof is a deductive argument to ensure the truth of a mathematical proposition. Other previously established statements, such as theorems or initial statements or axioms, may be used in argumentation. In principle, a proof can be traced back to generally accepted statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must show that a statement is always true (occasionally by listing all possible cases and showing that it is valid in each), rather than listing many confirmatory cases. An unproven statement that is believed to be true is known as a conjecture.

Proofs employ logic but usually include a good deal of natural language, which usually admits of some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered applications of rigorous informal logic. Purely formal proofs, written in symbolic language rather than natural language, are considered proof theory. The distinction between formal and informal proofs has led to examination of current and historical mathematical logic, mathematical quasi-empiricism, and mathematical formalism. The philosophy of mathematics is concerned with the role of language and logic in proofs, and in mathematics as language.

The fact of not knowing any proof of a theorem does not imply that it is not true; only the proof of the negation of this result implies that it is false.

Etymology and history

The word "test" comes from the Latin probare, which means 'to test'. Related modern words are the Spanish words "probar" ('to taste', 'to smell' or 'to try'), "probity", "probo" (or "proba") and "probabilidad", the German word probieren ('to try'), the Italian probare ('to try') and the English words probe and probation. The early use of the English term probity meant 'presentation of legal evidence'. A person of authority - which in general was anyone with a lot of money - was said to be a "proven" person, and his evidence outweighed any other testimony or empirical demonstration.

Plausibility arguments using heuristic devices such as images and analogies preceded strict mathematical proof. It is likely that the idea of proving a conclusion first appeared in connection with geometry, which originally meant 'measure of the land' or surveying. The development of mathematical proof is the primary product of ancient Greek mathematics, and one of its greatest achievements. Thales of Miletus (624-546 BC) proved some theorems in geometry. Eudoxus (408-355 BC) and Theaetetus (417-369 BC) formulated theorems but did not prove them. Aristotle (384-322 BC) said that definitions should describe the concept to be defined in terms of other already known concepts. Proofs in mathematics were revolutionized by Euclid (300 BC), who introduced the axiomatic method that is still used today, starting with undefined terms and axioms (propositions concerning undefined terms assumed to be evidently true, come from the Greek axios, meaning 'valuable'), and used these to prove theorems using deductive logic. His book, The Elements, was read by anyone who considered themselves educated in the West until the middle of the XX century. In addition to familiar theorems in geometry, such as the Pythagorean theorem, items include a proof that the square root of two is irrational and that there are infinitely many prime numbers.

Later advances took place in medieval Islamic mathematics. While the early Greek proofs were mostly geometric proofs, the development of arithmetic and algebra by Islamic mathematicians allowed for more general proofs that did not depend on geometry. In the X century AD. C., the Iraqi mathematician Al-Hashim gave to provide general proofs for numbers (rather than geometric proofs) when considering multiplication and division among others "by lines". He used this method to provide a proof of the existence of irrational numbers.

An inductive proof for arithmetic sequences was introduced in the Al-Fakhri (1000 AD) by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle. Alhazen also developed the method of proof by contradiction, as the first attempt to prove Euclidean's parallel postulate.

Modern proof theory treats proofs as inductively defined data structures. Axioms are no longer assumed to be "true" in any sense; this allows parallel mathematical theories to be created on alternate sets of axioms (see Axiomatic set theory and non-Euclidean geometry for examples).

Nature and purpose

As said, a proof is written in natural language, this being a rigorous argument with the purpose of convincing the audience of the veracity of a statement or definition. Standard rigor is not absolute and has varied throughout history. A demo can be presented in different ways depending on the expected audience. In order to gain acceptance, a demo has to meet common standards of rigor; an argument considered vague or incomplete must be rejected.

The concept of a proof is formalized in the field of mathematical logic. A formal proof is written in formal language rather than natural language. A formal proof is defined as a sequence of formulas in a formal language in which each formula is a logical consequence of the preceding ones. Having a formal proof definition makes the concept of a proof fun to study. In fact, the field of proof theory studies formal proofs and their properties, for example, the property of a statement having a formal proof. One application of the theory of proofs is to show that certain undecidable statements cannot have a proof.

The definition of a formal proof is supposed to capture the concept of a proof as it is written in the practice of mathematics. The soundness of this definition rests on the belief that a published proof can, in principle, be converted into a formal proof. However, outside of the realm of automated demo attendants, this is rarely done in practice. A classic question in philosophy asks whether mathematical proofs are analytical or synthetic. Kant, who introduced the distinction between analytic and synthetic, believed that proofs in mathematics are synthetic.

Proofs can be seen as aesthetic objects, admired for their mathematical beauty. The mathematician Paul Erdős described the proofs that he considered particularly elegant as coming from The Book, a hypothetical text that supposedly contains the most beautiful methods of proving each theorem. The essay The Proofs of “The Book” , published in 2009, presents 32 proofs that the editors of it find particularly satisfying.

Demo methods

Although in general there is no single thesis proof procedure, there are different types of proofs that are commonly used in mathematics:

• Demonstration by contrast (formalized and used in the syllogisms by Aristotle).
• Demonstration by reduction to the absurd (formalized and used by Aristotle) and, as a particular case, infinite decline
• Mathematical Induction
• Strong induction

On the other hand, despite the high degree of human intervention necessary to make a proof, there are also computational techniques that allow making automatic proofs, notably in the field of Euclidean geometry.

Direct proof

A proposition is stated, in the form “if p, then q”, where p is called the hypothesis (sufficient condition) and q is called the thesis or conclusion (necessary condition). For example, if it rains, the track is wet; this is that a sufficient condition for the track to get wet is that it rains. And if it rains, the track necessarily gets wet. In the mathematical context, from the truth of the hypothesis one arrives at the truth of the conclusion, using propositions whose certainty is previously known.

In direct proof, the conclusion is established by logically combining the previous axioms, definitions, and theorems. For example, direct proof can be used to establish that the sum of two even integers is always even:

Consider two integer pairs x e and. As pairs, they can be written as x= 2a e and= 2b, respectively, for integers a and b. Then the sum x+and= 2a+ 2b= 2(a+b). So. x+and has a factor of 2 and, by definition, it's pair. Therefore the sum of two integer pairs is pair.

This proof uses the definition of even integers, the properties of integers for closure under addition and multiplication, and distributivity.

Also a theorem can be enunciated in the form "p yes and only if q", which carries two statements "if..., then". It proves “if p..., then q” and “if q..., then p”. As an example, ${displaystyle a}$ It's an odd number if and only if ${displaystyle a+1}$ It's a pair. Induced of this kind, in practice, both can be directly demonstrated or by a reduction to the absurd. The important thing is the biconditional link.

Proof by the principle of mathematical induction

Mathematical induction is not a form of inductive reasoning. In a proof by mathematical induction, one proves a single "base case" and also a "rule of induction," which states that a certain case implies the next. Applying the rule of induction repeatedly, starting from the independently proven base case, proves many, sometimes infinitely many, other cases. Since the base case is true, infinity of the other cases must also be true, even if they cannot all be true. be directly tested given its infinity. An induction subset is infinitely descending. Infinite descent can be used to prove the irrationality of the square root of two.

A common application of mathematical induction is to prove that a property known to hold for a number holds for all natural numbers:

Be N = {1,2,3,4,...} the set of natural numbers, and P(n) the mathematical affirmation that involves the natural number n that belongs to N such that:

• (i) P(1) is true, p.e., P(n) is true for n = 1.

• (ii) P(n+1) is true wherever P(n) is true, p.e., P(n) is true implies that P(n+1) is true.

• Therefore P(n) is true for all natural numbers n.

For example, we can prove by induction that all integers in the form 2n + 1 are odd:

• (i) For n = 1, 2n + 1 = 2(1) + 1 = 3, and 3 is odd. Then P(1) is true.

• (ii) For 2n + 1 for some n, 2(n+1) + 1 = (2n+1) + 2. If 2n + 1 is odd, then (2n+1) + 2 must be odd, because adding 2 to an odd number gives an odd number. So P(n+1) is true if P(n) is true.

Therefore 2n + 1 is odd, for all natural numbers n.

It is common to say "proof by induction" instead of "proof by mathematical induction".

Proof by contraposition

The demonstration by contrast infers the conclusion "if the event p implies the event q, then no event q implies no event p», or mathematically: ${displaystyle (pRightarrow q)Rightarrow (no;qRightarrow no;p}$The statement «if not q, Then no. p» is called the counterpositive of the affirmation of “si p, then. q».

A non-mathematical logical example could be the following: let's imagine that a restaurant offers paella on its menu every Thursday. That is, the event "Thursday" implies the event "paella". It may be that we go on a Monday and there is paella, or it may be that we go on a Tuesday and there is not, but what we know for sure is that every Thursday there is paella. Of all the possible logical conclusions that derive from the previous statement, only one of them is true: that if we go one day and there is no paella, then surely it is not Thursday. Or put another way, "no paella" implies "no Thursday."

A mathematical example: contraposition can be used to state that if a² is odd, then a is odd. It is evident that a even implies a² even (if we multiply an even number by itself, we get another even number). Therefore, we can say that if a² is not even, then a is not even either. Or put another way, if a² is odd, then a is odd.

In the system of real numbers you have the theorem «si ${displaystyle ab=0}$, then ${displaystyle a=0}$ or ${displaystyle b=0}$», which carries the counterpositive proposition «sisitive ${displaystyle aneq 0}$ and ${displaystyle bneq 0}$ then. ${displaystyle abneq 0}$».

Proof by reduction to absurdity

In the demonstration by contradiction (also known as reductio ad absurdum, which means ‘by reduction to the absurd’ in Latin), it is shown that if a certain assertion is true, a logical contradiction occurs, therefore that assertion is false. A famous example of demonstration by contradiction shows that ${displaystyle {sqrt {2}}}$ It's an irrational number:

Supposing that ${displaystyle {sqrt {2}}}$ is a rational number, so by definition ${displaystyle {sqrt {2}}={a over b}}}$ where a and b are two different integers of zero without common factors. Therefore, ${displaystyle {{sqrt {2}b}=a}$. Lifting the square on both sides has to ${displaystyle 2b^{2}=a^{2}}$. As 2 divides to the left side, it must also divide to the right side (otherwise, a number equals a non number). So, ${displaystyle a^{2}$ it's a pair, which implies that ${displaystyle a}$ Must be a pair too. So we can write ${displaystyle a=2c}$where c is also integer. Substituting in the original equation we have ${displaystyle 2b^{2}=(2c)^{2}=4c^{2}$. Splitting on both sides by 2 we have ${displaystyle b^{2}=2c^{2}}$. But then, by the same argument as before, 2 divides to ${displaystyle b^{2}}$, then b must be pair. However, if a and b are both integers, they share a factor, which is 2. This contradicts our assumption, so we are forced to conclude that ${displaystyle {sqrt {2}}}$ It's an irrational number.

Constructive demonstration or by construction

Proof by construction, or proof by example, is the construction of a concrete example with a specific property to show that something possessing that property exists. Joseph Liouville, for example, proved the existence of transcendental numbers by constructing an explicit example. It can also be used to construct a counterexample to prove that the proposition that all elements have a certain property is not true.

This form of proof was applied by Cantor to prove that the set of real numbers is uncountable. The demonstrative scheme starts from the hypothesis that all real numbers can be enumerated and arranged in a sequence, and a real number that does not appear in such a sequence is then constructed. There is a contradiction with the initial hypothesis that assumed that all the real numbers were included in the sequence. Hence the hypothesis of the enumeration of real numbers is absurd; so that the opposite hypothesis, that is, Cantor's proposition that the set of real numbers is not countable, is proven.

Proof for completeness

In the proof by completeness, the conclusion is established by dividing it into a finite number of cases and testing each one separately. The number of cases can sometimes be very large. For example, the first proof of the four color theorem was a completeness proof with 1936 cases. This demonstration was controversial as most of the cases were verified with a computer program and not by hand. The shortest known proof of the four color theorem was from 2011 and still has more than 600 instances.

Probabilistic Proof

A probabilistic proof is one in which an instance is shown to exist, with certainty, using methods of probability theory. This is not to be confused with an argument that a theorem is "probably" TRUE. This type of reasoning can be called a "plausibility argument" and does not entail a proof. In the case of the Collatz conjecture it is clear how far this is from being a genuine proof. Probabilistic proof, like proof by construction, is one of many ways to prove existence theorems.

Proof by Combinatorics

A combinatorics proof establishes the equivalence of different expressions by showing that they count for the same object in different ways. A bijection between two sets is often used to show that the expressions for their two sizes are equal. Alternatively, a double count argument provides two different expressions for the size of a single set, again showing that the two expressions are equal.

Non-constructive proof

A non-constructive proof states that a mathematical object with a certain property exists without explaining how such an object can be found. Often these take the form of a proof by contradiction (reduction to absurdity) proving that if a proposition were not true, then it would lead to a contradiction. In contrast, a constructive proof establishes that a particular object exists by providing a method to find it.

A famous non-constructive demonstration example shows that there are two irrational numbers a and b such as ${displaystyle a^{b}$ is a rational number:

Or ${displaystyle {sqrt {2}}{sqrt {2}}}}}$ It's a rational number and we're done. ${displaystyle a=b={sqrt {2}}}$), or ${displaystyle {sqrt {2}}{sqrt {2}}}}}$ It's irrational so we can write ${displaystyle a={sqrt {2}}{sqrt {2}}}}$ and ${displaystyle b={sqrt {2}}}}$. This produces ${displaystyle left({sqrt {2}}{sqrt {2}}{right)^{sqrt {2}}={sqrt {2}}{2}}{2}=2}}}$, which is a therefore rational form ${displaystyle a^{b}$.

Statistical tests in pure mathematics

The expression "statistical proof" can be used technically or colloquially in areas of pure mathematics, such as those involving cryptography, chaotic series, and probabilistic or analytic number theory. It is not as commonly used to refer to a mathematical proof in the area of mathematics known as mathematical statistics. See also the “statistical proof using data” section below.

Computer Aided Testing

Until the 20th century it was assumed that any proof should, in principle, be reviewed by a competent mathematician to confirm its validity. In any case, computers are now used to prove theorems and to make calculations that would take a long time for a human or group of them to review; The first proof of the four color theorem is an example of a computer aided proof. Some mathematicians are concerned that the possibility of an error in a computer program or an execution error in its calculations may affect the validity of such computer-assisted proofs. In practice the chances of an error invalidating a computer-aided proof can be reduced by incorporating redundancy and self-checks into the calculations, and by developing multiple, independent approaches and programs. Errors cannot be fully overcome in case of human verification of a proof either, especially if the proof contains natural language and requires deep mathematical background.

Undecidable Statements

A statement that is neither positively nor negatively provable from a set of axioms is called undecidable (from those axioms). An example is the parallel postulate, which is neither provable nor refutable from the other axioms of Euclidean geometry.

Mathematicians have shown that there are many statements that are neither provable nor refutable in Zermelo-Fraenkel set theory with the axiom of choice (ZFC), the standard system of set theory in mathematics (assuming ZFC is consistent); see the list of undecidable statements in ZFC.

Gödel's first incompleteness theorem shows that many axiomatic systems of mathematical interest will have undecidable sentences.

Heuristic and experimental mathematics

While early mathematicians such as Eudoxus of Cnidus did not use proofs, from Euclid to the founding developments of mathematics in the late century XIX and XX, proofs became an essential part of mathematics. With the increase in computational power in the 1960s, significant work began to be done in experimental mathematics investigating mathematical objects outside the proof-theorem framework. The early pioneers of those methods intended that the work would eventually be translated into the classical proof-theorem framework, for example, the early development of fractal geometry, which was ultimately highly appreciated.

Related concepts

Visual demonstration

Although not a formal proof, a visual proof of a mathematical theorem is sometimes called a "wordless proof". The image on the left shown below is an example of the historical visual proof of the Pythagorean Theorem in the case of the triangle of sides with measures (3,4,5).

• 

  Visual demonstration for the triangle of sides (3, 4, 5) as shown in the Chou Pei Suan Ching 500–200 BC.

• 

  Animated visual demonstration of the theorem of Pythagoras for retrofitting.

• 

  A second animated demonstration of the theorem of Pythagoras.

Elementary proof

An elementary demo is a demo that uses only basic techniques. The term is used more specifically in number theory to refer to proofs that do not make use of complex analysis. For some time it was thought that certain theorems, such as the prime number theorem, could only be proved using "higher mathematics". However, as time passed, many of these results could be reproven using only elementary techniques.

Two Column Demo

A demonstration of two columns published in 1913

A particular way of organizing a proof, using two parallel columns, is often used in elementary geometry classes in the US. The proof is written as a series of lines in two columns. In each line, the left column contains a proposition, while the right column contains a short explanation of how the corresponding proposition in the left column is either an axiom, a hypothesis, or can be logically derived from the preceding propositions. The left column is typically labeled "Claims" and the right, "Reasons".

Colloquial use of n#34;mathematical proof#34;

The expression «mathematical proof» is popularly used to refer to using mathematical methods or discussing based on mathematical objects ―such as numbers―, to prove something of daily life, or when the data used in a discussion are numerical. It also sometimes means "statistical proof" (more below), especially when used to argue with data.

Statistical proof using data

"Statistical proof" using data refers to the application of statistics, data analysis, or Bayesian analysis to infer propositions concerning the probability of data. While a mathematical proof is used to establish theorems in statistics, these are usually not mathematical proofs in the sense that the assumptions from which statements in probability are derived require empirical evidence. outside of mathematics to be verified. In physics, in addition to statistical methods, "statistical proof" can refer to specialized mathematical methods of physics applied to analyze data in experiments in particle physics or observational studies in cosmology. "Statistical proof" can also mean the raw data or a compelling diagram that uses data, such as scatter plots, where the data or diagram is adequately convincing without further analysis.

Proofs of Inductive Logic and Bayesian Analysis

Proofs using inductive logic, while considered mathematical in nature, seek to establish proportions with a degree of certainty, which acts similarly to probability, and could be less than a certainty. Bayesian analysis establishes assertions such as the degree of subjective belief of the person. Inductive logic should not be confused with mathematical induction.

Tests as mental objects

Psychologism sees mathematical proofs as psychological or mental objects. Philosophers of mathematics, such as Leibniz, Gottlob Frege, and Rudolf Carnap, attempted to develop a semantics for what they considered to be the language of thought, where the standards of mathematical proof could be applied to empirical science.

Influences of mathematical proof methods outside mathematics

Philosophers and mathematicians such as Baruch Spinoza attempted to formulate philosophical arguments in an axiomatic style, where the standards of mathematical proof could be applied in general philosophical argument. Other philosophers and mathematicians attempted to use the standards of mathematical proof and reason, without empiricism, such as Descartes' cogito argument.

End of a demo

Sometimes, the abbreviation QED is written to indicate the end of a demo. This abbreviation means quod erat demonstraterandum, which in Latin means 'what was wanted to be demonstrated'. A more common alternative is to use a square or rectangle, such as □ or ∎, known as tombstone or halmos (after eponymous Paul Halmos). Often, "what you wanted to prove" is written verbally by writing QED, □, or ∎ in an oral presentation on a blackboard.

Additional information

Example of a proof by contradiction Also called demonstration of the absurd

Demonstration of the assertion ${displaystyle 1 20050,}$

Before demonstrating this, we must be clear that there are certain axioms that will allow us, in this case, to prove our statement. Since we will be relying on axioms, we have that our proof (with each logical step being correct) is true.

• We will use the following axioms of the actual numbers:

Ax1. ${displaystyle 1not =0,}$

Ax2. Yeah. ${displaystyle a vocabularyb,}$ and ${displaystyle vs.$With real a,b,c. Then ${displaystyle ac ingredientbc,}$

Assuming certain of these axioms we can begin with our demonstration. Suppose for a moment, contrary to what is expected, ${displaystyle 1 tax0,}$ and let us see that we come to a contradiction. Post

${displaystyle 1 tax0,}$applying axiom Ax2 by multiplying by 1 (which is less than zero), we have to

${displaystyle 1 20050,}$which is a contradiction.

As our hypothesis was that ${displaystyle 1 tax0,}$And this is fake, all we can say now is ${displaystyle 0leq 1}$. But the axiom Ax1 says the only possibility where there is no contradiction is that effectively ${displaystyle 1not =0,}$

Then

${displaystyle 1 20050,}$

Reasoning

It is clear that what we must have is a contradiction. To do this, we must first formulate a hypothesis, and check if it is true or not. If it is not, it will lead us to a contradiction. It should be clear that our hypothesis begins when it is said that 1 is less than zero and not in the axioms mentioned above (because these are already proven or assumed to be true and therefore do not require further analysis). Since we know that the statement "'1 is less than zero" is false, we should arrive at a contradiction. But it is not enough just to know it, since it must be demonstrated.

Our hypothesis was that one was less than zero and, after some correct logical steps using the axioms, we concluded that one was greater than zero, which clearly cannot be true, since by the law of trichotomy, two numbers must fulfill one and only one of the following relations

${displaystyle x parenty,}$; ${displaystyle x visy,}$ or ${displaystyle x=y,}$,

but never two or three together. Then, since our hypothesis leads us to a contradiction, it is false, and we must consider all the possibilities, except that one. That is: since one is not less than zero, it must, necessarily, be greater than or equal to this (zero). But the axiom first says that one is non-zero, so there is only the option of 1 being greater than zero.

Incorrect reasoning

A common mistake among those who begin the study of these matters is to think that they have reached a contradiction without having done so. For example: They assume that

${displaystyle 1 tax0,}$then adding (-1) to both sides

${displaystyle},$ which is a contradiction since ${displaystyle 0 vocabulary{-1},}$.

This reasoning has a mistake, as we do not come to a contradiction. Our hypothesis was that 1 was less than zero and therefore with the procedures performed ${displaystyle 1 ingredient},$That's true. In this case, the statement ${displaystyle 0 vocabulary{-1},}$ It's fake.

Note that to arrive at a contradiction we must have the following:

• A P statement (in our hypothesis) that says this is true.
• A conclusion that says P is false.

Clearly, no statement can do this. In logic, this statement would be:

P is true and ~P is true, which reads "P is true and P is not true."

Example of a proof by induction

Prove that

${displaystyle sum _{k=1}{n}{frac {1}{k(k+1)}}}}{frac {n}{n+1}}}}}}}}$

Demo

• We must check if the claim is true to ${displaystyle {mathit {n=1},!}$, since the summary part from ${displaystyle {mathit {k=1},!}$.

Sea ${displaystyle {mathit {n=1},!}$, then

${displaystyle sum _{k=1}{1}{frac {1}{k(k+1)}}}{frac {1}{1+1}}}{frac {1}{1}{2}}}}}}}$.

and the affirmation is true for ${displaystyle {mathit {n=1},!}$.

• Suppose now, that the claim is true for a ${displaystyle {mathit {native1},!}$ fixed, and let's see what happens to ${displaystyle {mathit {n+1},!}$.

By property of summations we have that

${displaystyle sum _{k=1}^{n+1}{frac {1}{k(k+1)}}}=sum _{k=1}{n}{frac {1}{k(k+1)}}}{k={k=n+1}{^{n+1}{frac {1}{k(k(k+1)}}}}{{{{{{{{{k=1}}}{k={k=1}}}}{k}{x1}}}}}{k={x1}}}}}{x1}}}{x1}}}}{x1}}{x1}}{x1}{x1}}{x1}}{x1}}}}}}{x1}{x1}}}}{x1}{x1}}}}{x1}}}{x1}}}}{k(k(k(k(k(k(k(k(k(k(k(k(k(k !$

as the claim is true to ${displaystyle {mathit {n},!}$We have to

${displaystyle sum _{k=1}^{n+1}{frac {1}{k(k+1)}}}={frac {n}{n+1}}}{frac {1}{(n+1)}{(n+2)}}}$

ordering

${displaystyle sum _{k=1}^{n+1}{frac {1}{k(k+1)}}}{frac {n(n+2)+1}{(n+1)}{(n+2)}}}}}}$

${displaystyle sum _{k=1}^{n+1}{frac {1}{k(k+1)}}}}={frac {n^{2}+2n+1}{(n+1)}}}{(n+2)}}}}}}}$

${displaystyle sum _{k=1}^{n+1}{frac {1}{k(k+1)}}}}{frac {(n+1)^{2}{(n+1)}{(n+2)}}}}}{#$

Like ${displaystyle {mathit {n},!}$ It's different from ${displaystyle {mathit {(-1)},!}$, we can simplify and

${displaystyle sum _{k=1}^{n+1}{frac {1}{k(k+1)}}}{frac {(n+1)}{(n+1)+1}}}}}}}}{displaystyle sum$

which is what we wanted to demonstrate.

Thus, the affirmation is also true for ${displaystyle {mathit {n+1},!}$.

Then the affirmation is true for everything ${displaystyle nin mathbb {N} }$.

Reasoning

The Induction principle says that given an assertion ${displaystyle {mathit {P(n)}},!}$This is true only if it is fulfilled

• ${displaystyle {mathit {P(1)},!}$ That's true.
• Yeah. ${displaystyle {mathit {P(n)}},!}$ That's true, then. ${displaystyle {mathit {P(n+1)},!}$ It is too.

So, like ours ${displaystyle {mathit {P(n)}},!}$ is what we want to prove, we must see if it is true his first term. In this case, ${displaystyle {mathit {n=1},!}$.

Note that we should not necessarily have 1 left in the summary. What indicates to us ${displaystyle {mathit {P(1)},!}$ is that we should see if it is true for the first term. As the statement was fulfilled ${displaystyle {mathit {n=1},!}$the next step was to see if ${displaystyle {mathit {n},!}$It was fulfilled ${displaystyle {mathit {n+1},!}$. So then, we use what we want to see if it is true in the only left member of our equation. Then applying properties of the summation, we can decompose the summation in parts and leave what we know is true, separated from what can be applied by definition. We know it's true. ${displaystyle {mathit {n},!}$, therefore the first member of the right side can be replaced, while the second member only applies the definition of summation.

Then, adding the fractions and grouping, we conclude that the first member on the left side of our equation can be expressed as what we want to demonstrate. Therefore, we conclude that the assertion is true for ${displaystyle {mathit {n+1},!}$.

Formal definition In mathematical logic and propositional logic, a proof is a finite sequence of well-formed logical formulas:

${displaystyle F_{1},dotsF_{n}}$

such that each F_i is either an axiom or a theorem that follows from two previous formulas F_j and F_k (such that j<i and k<i) by a rule of valid deduction. That is to say,

${displaystyle forall F_{i}:(F_{i} {mbox{es un axioma}}}})lor (exists F_{j},F_{k}: F_{i} {mbox{se deduce de}}}}{ F_{j},F_{k}{mbox{con}}} j,k$

Given a proof like the one above, if the final element F_n is not an axiom then it is a theorem. From the point of view of formal languages, the set of provable theorems coincides with the set of syntactically well-formed sequences of well-formed formulas.

Fonts

• Pólya, G. (1954), Mathematics and Plausible Reasoning, Princeton University Press..
• Fallis, Don (2002), "What Do Mathematicians Want? Probabilistic Proofs and the Epistemic Goals of Mathematicians”, Logique et Analyse 45: 373-388..
• Franklin, J.; Daoud, A. (2011), Proof in Mathematics: An Introduction, Kew Books, ISBN 0-646-54509-4..
• Solow, D. (2004), How to Read and Do Proofs: An Introduction to Mathematical Thought ProcessesWiley, ISBN 0-471-68058-3..
• Velleman, D. (2006), How to Prove It: A Structured Approach, Cambridge University Press, ISBN 0-521-67599-5..