Mathematical proof - Citizendia

In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true, within the accepted standards of the field. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence A proof is a logically deduced argument, not an empirical one. Deductive reasoning is Reasoning which uses deductive Arguments to move from given statements ( Premises to Conclusions which must be true if the A central concept in Science and the Scientific method is that all Evidence must be empirical, or empirically based that is dependent on evidence That is, the proof must demonstrate that a proposition is true in all cases to which it applies, without a single exception. An unproven proposition believed or strongly suspected to be true is known as a conjecture. In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of

Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. Logic is the study of the principles of valid demonstration and Inference. In the Philosophy of language, a natural language (or ordinary language) is a Language that is spoken or written in phonemic-alphabetic or phonemically-related In fact, the vast majority of proofs in written mathematics can be considered as applications of informal logic. Informal logic (or occasionally non-formal logic) is the study of arguments as presented in ordinary language as contrasted with the presentations of arguments in Purely formal proofs are considered in proof theory. see also Mathematical proof, Proof theory, and Axiomatic system. Proof theory is a branch of Mathematical logic that represents proofs as formal Mathematical objects facilitating their analysis by mathematical techniques The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). Proof theory is a branch of Mathematical logic that represents proofs as formal Mathematical objects facilitating their analysis by mathematical techniques Mathematical practice is used to distinguish the working practices of professional mathematicians ( e Quasi-empiricism in Mathematics is the attempt in the Philosophy of mathematics to direct philosophers' attention to Mathematical practice, in The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language. The philosophy of mathematics is the branch of Philosophy that studies the philosophical assumptions foundations and implications of Mathematics. The central question involved in discussing mathematics as a language can be stated as follows What do we mean when we talk about the language of mathematics

Regardless of one's attitude to formalism, the result that is proved to be true is a theorem; in a completely formal proof it would be the final line, and the complete proof shows how it follows from the axioms alone by the application of the rules of inference. In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject Once a theorem is proved, it can be used as the basis to prove further statements. A theorem may also be referred to as a lemma if it is used as a stepping stone in the proof of a theorem. In Mathematics, a lemma (plural lemmata or lemmas from the Greek λήμμα "lemma" meaning "anything which is received The axioms are those statements one cannot, or need not, prove. These were once the primary study of philosophers of mathematics. Today focus is more on practice, i. Mathematical practice is used to distinguish the working practices of professional mathematicians ( e e. acceptable techniques.

1 Methods of proof
2 End of a proof
3 See also
4 References
5 Sources
6 External links

Methods of proof

Direct proof

Main article: Direct proof

In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. In Mathematics and Logic, a direct proof is a way of showing the Truth or falsehood of a given statement by a straightforward combination of established For example, direct proof can be used to establish that the sum of two even integers is always even:

For any two even integers

x

and

y

we can write

x = 2 a

and

y = 2 b

for some integers

a

and

b

, since both

x

and

y

are multiples of 2. In Mathematics, the parity of an object states whether it is even or odd The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French But the sum

x + y = 2 a + 2 b = 2(a + b)

is also a multiple of 2, so it is therefore even by definition.

This proof uses definition of even integers, as well as distribution law. In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law

Proof by induction

Main article: Mathematical induction

In proof by induction, first a "base case" is proved, and then an "induction rule" is used to prove a (often infinite) series of other cases. Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness Since the base case is true, the infinity of other cases must also be true, even if all of them cannot be proved directly because of their infinite number. A subset of induction is Infinite descent. In Mathematics, a proof by infinite descent is a particular kind of proof by Mathematical induction. Infinite descent can be used to prove the irrationality of the square root of two. The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2} or √2

The principle of mathematical induction states that: Let N = { 1, 2, 3, 4, . . . } be the set of natural numbers and P(n) be a mathematical statement involving the natural number n belonging to N such that (i) P(1) is true, ie, P(n) is true for n = 1 (ii) P(m + 1) is true whenever P(m) is true, ie, P(m) is true implies that P(m + 1) is true. Then P(n) is true for all natural numbers n.

Proof by transposition

Main article: Transposition (logic)

Proof by Transposition establishes the conclusion "if p then q" by proving the equivalent contrapositive statement "if not q then not p". Transposition (mathematics -->In the methods of Deductive reasoning in Classical logic, " For contraposition in the field of traditional logic see Contraposition (traditional logic.

Proof by contradiction

Main article: Reductio ad absurdum

In proof by contradiction (also known as reductio ad absurdum, Latin for "reduction into the absurd"), it is shown that if some statement were false, a logical contradiction occurs, hence the statement must be true. Reductio ad absurdum ( Latin for "reduction to the absurd" also known as an apagogical argument, reductio ad impossibile This method is perhaps the most prevalent of mathematical proofs. A famous example of a proof by contradiction shows that $\sqrt{2}$ is irrational:

Suppose that $\sqrt{2}$ is rational, so $\sqrt{2} = {a\over b}$ where a and b are non-zero integers with no common factor (definition of rational number). In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction Thus, $b\sqrt{2} = a$ . Squaring both sides yields 2b² = a². Since 2 divides the left hand side, 2 must also divide the right hand side (as they are equal and both integers). So a² is even, which implies that a must also be even. So we can write a = 2c, where c is also an integer. Substitution into the original equation yields 2b² = (2c)² = 4c². Dividing both sides by 2 yields b² = 2c². But then, by the same argument as before, 2 divides b², so b must be even. However, if a and b are both even, they share a factor, namely 2. This contradicts our assumption, so we are forced to conclude that $\sqrt{2}$ is irrational.

Proof by construction

Main article: Proof by construction

Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists. In Mathematics, a constructive proof is a method of proof that demonstrates the existence of a Mathematical object with certain properties by creating Joseph Liouville, for instance, proved the existence of transcendental numbers by constructing an explicit example. Joseph Liouville ( March 24 1809 &ndash September 8 1882) was a French Mathematician. In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation In Number theory, a Liouville number is a Real number x with the property that for any positive Integer n, there exist integers

Proof by exhaustion

Main article: Proof by exhaustion

In Proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. Proof by exhaustion, also known as proof by cases, perfect induction, or the brute force method, is a method of Mathematical proof in which The number of cases sometimes can become very large. For example, the first proof of the four colour theorem was a proof by exhaustion with 1,936 cases. The four color theorem (also known as the four color map theorem) states that given any plane separated into regions such as a political map of the states of a country This proof was controversial because the majority of the cases were checked by a computer program, not by hand. The shortest known proof of the four colour theorem today still has over 600 cases.

Probabilistic proof

Main article: Probabilistic method

A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of probability theory. This article is not about Probabilistic algorithms which give the right answer with high probability but not with certainty nor about Monte Probability theory is the branch of Mathematics concerned with analysis of random phenomena This is not be confused with an argument that a theorem is 'probably' true. The latter type of reasoning can be called a 'plausibility argument' and is not a proof; in the case of the Collatz conjecture it is clear how far that is from a genuine proof. The Collatz conjecture is an unsolved Conjecture in Mathematics. ^[1] Probabilistic proof, like proof by construction, is one of many ways to show existence theorems. In Mathematics, an existence theorem is a theorem with a statement beginning 'there exist(s.

Combinatorial proof

Main article: Combinatorial proof

A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. The term combinatorial proof is often used in either of two senses A proof by double counting. Usually a bijection is used to show that the two interpretations give the same result. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

Nonconstructive proof

Main article: Nonconstructive proof

A nonconstructive proof establishes that a certain mathematical object must exist (e. g. "Some X satisfies f(X)"), without explaining how such an object can be found. Often, this takes the form of a proof by contradiction in which the nonexistence of the object is proven to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. A famous example of a nonconstructive proof shows that there exist two irrational numbers $a$ and $b$ such that $a b$ is a rational number:

Either $\sqrt{2}^{\sqrt{2}}$ is a rational number and we are done (take $a=b=\sqrt{2}$ ), or $\sqrt{2}^{\sqrt{2}}$ is irrational so we can write $a=\sqrt{2}^{\sqrt{2}}$ and $b=\sqrt{2}$ . In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions This then gives $\left (\sqrt{2}^{\sqrt{2}}\right )^{\sqrt{2}}=\sqrt{2}^{2}=2$ , which is thus a rational of the form

a b

Proof nor disproof

There is a class of mathematical statements for which neither a proof nor disproof exists, using only ZFC, the standard form of axiomatic set theory. Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common Examples include the continuum hypothesis; see further List of statements undecidable in ZFC. In Mathematics, the continuum hypothesis (abbreviated CH) is a Hypothesis, advanced by Georg Cantor, about the possible sizes of Infinite The following is a list of mathematical statements that are undecidable in ZFC (the Zermelo–Fraenkel axioms plus the Axiom of choice) assuming that ZFC Under the assumption that ZFC is consistent, the existence of such statements follows from Gödel's (first) incompleteness theorem. In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most Whether a particular unproven proposition can be proved or disproved using a standard set of axioms is not always obvious, and can be extremely technical to determine.

Elementary proof

Main article: Elementary proof

An elementary proof is (usually) a proof which does not use complex analysis. In Mathematics a proof is said to be elementary if it avoids difficult ideas from distant areas of mathematics For some time it was thought that certain theorems, like the prime number theorem, could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques.

End of a proof

Main article: Q.E.D.

Sometimes, the abbreviation "Q. QED is an abbreviation of the Latin phrase "la '''quod erat demonstrandum'''" which means literally "that which was to be demonstrated" E. D. " is written to indicate the end of a proof. This abbreviation stands for "Quod Erat Demonstrandum", which is Latin for "that which was to be demonstrated". Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. An alternative is to use a square or a rectangle, such as □ or ∎, known as a "tombstone" or "halmos". The tombstone, halmos, or end of proof mark "□" is used in Mathematics to denote the end of a proof in place of the traditional abbreviation

References

^ While most mathematicians do not think that probabilistic evidence ever counts as a genuine mathematical proof, a few mathematicians and philosophers have argued that at least some types of probabilistic evidence (such as Rabin’s probabilistic algorithm for testing primality) are as good as genuine mathematical proofs. Proof theory is a branch of Mathematical logic that represents proofs as formal Mathematical objects facilitating their analysis by mathematical techniques In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models A computer-assisted proof is a Mathematical proof that has been at least partially generated by computer Automated theorem proving ( ATP) or automated deduction, currently the most well-developed subfield of Automated reasoning (AR is the In Mathematics, there are a variety of spurious proofs of obvious Contradictions Although the proofs are flawed the errors usually by design are comparatively subtle A list of articles with Mathematical proofs Theorems of which articles are primarily devoted to proving them See also:CategoryArticle proofs Proofs from THE BOOK is a book of Mathematical proofs by Martin Aigner and Günter M Proof by intimidation is a jocular term used mainly in Mathematics to refer to Mathematical proofs which are so complex so long-winded and so poorly presented by the A randomized algorithm or probabilistic algorithm is an Algorithm which employs a degree of randomness as part of its logic See, for example, Davis, Philip J. (1972), "Fidelity in Mathematical Discourse: Is One and One Really Two?" American Mathematical Monthly 79:252-63. Fallis, Don (1997), "The Epistemic Status of Probabilistic Proof. " Journal of Philosophy 94:165-86.

Sources

Polya, G. Mathematics and Plausible Reasoning. Princeton University Press, 1954.
Fallis, Don (2002) “What Do Mathematicians Want? Probabilistic Proofs and the Epistemic Goals of Mathematicians.” Logique et Analyse 45:373-88.
Franklin, J. and Daoud, A. Proof in Mathematics: An Introduction. Quakers Hill Press, 1996. ISBN 1-876192-00-3
Solow, D. How to Read and Do Proofs: An Introduction to Mathematical Thought Processes. Wiley, 2004. ISBN 0-471-68058-3
Velleman, D. How to Prove It: A Structured Approach. Cambridge University Press, 2006. ISBN 0-521-67599-5

External links

What are mathematical proofs and why they are important?
How To Write Proofs by Larry W. Cusick
How to Write a Proof by Leslie Lamport, and the motivation of proposing such a hierarchical proof style. Dr Leslie Lamport (born February 7, 1941 in New York City) is an American computer scientist.
Proofs in Mathematics: Simple, Charming and Fallacious
The Seventeen Provers of the World, ed. by Freek Wiedijk, foreword by Dana S. Scott, Lecture Notes in Computer Science 3600, Springer, 2006, ISBN 3-540-30704-4. Contains formalized versions of the proof that $\sqrt{2}$ is irrational in several automated proof systems.
What is Proof? Thoughts on proofs and proving.
ProofWiki.org The online compendium of mathematical proofs.

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