# Mathematical proof

**proofproofsproveprovenprovedprovingmathematical proofsdemonstrationformal proofproof method**

A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.wikipedia

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### Theorem

**theoremspropositionconverse**

The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference.

In mathematics, a theorem is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms, or on the basis previously established statements such as other theorems.

### Proof theory

**proof-theoreticProof-theoreticallyderive**

Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. 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 major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques.

### Philosophy of mathematics

**mathematical realismmathematical Platonismmathematics**

The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

Notions of axiom, proposition and proof, as well as the notion of a proposition being true of a mathematical object (see Assignment (mathematical logic)), were formalized, allowing them to be treated mathematically.

### Mathematical practice

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).

Mathematical practice comprises the working practices of professional mathematicians: selecting theorems to prove, using informal notations to persuade themselves and others that various steps in the final proof are convincing, and seeking peer review and publication, as opposed to the end result of proven and published theorems.

### Euclid's Elements

**ElementsEuclid's ''ElementsEuclid**

His book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century.

It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions.

### Mathematical induction

**inductioninductivelyproof by induction**

An inductive proof for arithmetic sequences was introduced in the Al-Fakhri (1000) by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle.

Mathematical induction is a mathematical proof technique.

### Axiomatic system

**axiomatizationaxiomatic methodaxiom system**

Mathematical proofs were revolutionized by Euclid (300 BCE), who introduced the axiomatic method still in use today, starting with undefined terms and axioms (propositions regarding the undefined terms assumed to be self-evidently true from the Greek "axios" meaning "something worthy"), and used these to prove theorems using deductive logic.

A formal proof is a complete rendition of a mathematical proof within a formal system.

### Argument-deduction-proof distinctions

**argument**

A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.

A proof of the Pythagorean theorem is a deduction that might use several premises — axioms, postulates, and definitions — and contain dozens of intermediate steps.

### Proof by contradiction

**contradictionby contradictionindirect proof**

Alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate.

In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction.

### Euclidean geometry

**plane geometryEuclideanEuclidean plane geometry**

Alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate.

The Elements begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of formal proof.

### Binomial theorem

**binomial expansionbinomial formulabinomial**

An inductive proof for arithmetic sequences was introduced in the Al-Fakhri (1000) by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle.

Al-Karaji described the triangular pattern of the binomial coefficients and also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using an early form of mathematical induction.

### Mathematical beauty

**mathematical eleganceelegantAesthetics**

Proofs may be viewed as aesthetic objects, admired for their mathematical beauty.

Mathematicians describe an especially pleasing method of proof as elegant.

### Euclid

**Euclid of AlexandriaEuklidGreek Mathematician**

Mathematical proofs were revolutionized by Euclid (300 BCE), who introduced the axiomatic method still in use today, starting with undefined terms and axioms (propositions regarding the undefined terms assumed to be self-evidently true from the Greek "axios" meaning "something worthy"), and used these to prove theorems using deductive logic.

Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.

### Proofs from THE BOOK

**maintains the most elegant proofs of mathematical theorems**

The book Proofs from THE BOOK, published in 2003, is devoted to presenting 32 proofs its editors find particularly pleasing.

Proofs from THE BOOK is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler.

### Mathematics in medieval Islam

**mathematicianmathematicsIslamic mathematics**

Further advances also took place in medieval Islamic mathematics.

In between, implicit proof by induction for arithmetic sequences was introduced by al-Karaji (c.

### Pascal's triangle

**Pascal triangleJia Xian triangleKhayyam triangle**

An inductive proof for arithmetic sequences was introduced in the Al-Fakhri (1000) by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle.

It is not difficult to turn this argument into a proof (by mathematical induction) of the binomial theorem.

### Conjecture

**conjecturalconjecturesconjectured**

An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if the proposition is frequently used as an assumption to build upon similar mathematical work.

There are various methods of doing so; see methods of mathematical proof for more details.

### Bijective proof

**bijectivebijectionbijective approach**

Often a bijection between two sets is used to show that the expressions for their two sizes are equal.

In combinatorics, bijective proof is a proof technique that finds a bijective function (that is, a one-to-one and onto function)

### Formal proof

**prooflogical prooflogical proofs**

Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory.

### Counterexample

**counter-examplecounterexamples**

It can also be used to construct a counterexample to disprove a proposition that all elements have a certain property.

In this case, she can either attempt to prove the truth of the statement using deductive reasoning, or she can attempt to find a counterexample of the statement if she suspects it to be false.

### Mathematical object

**mathematical objectsobjectsgeometric object**

A nonconstructive proof establishes that a mathematical object with a certain property exists — without explaining how such an object are to be found.

In mathematical practice, an object is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs.

### Proof by contrapositive

**contrapositivelyProof by contrapositionproof by contraposition (contrapositive)**

Proof by contraposition infers the statement "if p then q" by establishing the logically equivalent contrapositive statement: "if not q then not p".

In mathematics, proof by contrapositive, or proof by contraposition, is a rule of inference used in proofs, where one infers a conditional statement from its contrapositive.

### Proof without words

**visual proofA proof without words of Jensen's inequalitygraphical visualization**

Although not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a "proof without words".

In mathematics, a proof without words is a proof of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text.

### Parity (mathematics)

**even numberodd numberparity**

For example, direct proof can be used to prove that the sum of two even integers is always even:

Modern computer calculations have shown this conjecture to be true for integers up to at least 4 × 10 18, but still no general proof has been found.

### Tombstone (typography)

**tombstone ∎ end of proof**

A more common alternative is to use a square or a rectangle, such as □ or ∎, known as a "tombstone" or "halmos" after its eponym Paul Halmos.

In mathematics, the tombstone, halmos, end of proof, or Q.E.D. mark "∎" (or "□") is a symbol used to denote the end of a proof, in place of the more traditional abbreviation "Q.E.D."