Double counting (proof technique)
In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting t
Minimal counterexample
In mathematics, a minimal counterexample is the smallest example which falsifies a claim, and a proof by minimal counterexample is a method of proof which combines the use of a minimal counterexample
Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the
Combinatorial proof
In mathematics, the term combinatorial proof is often used to mean either of two types of mathematical proof:
* A proof by double counting. A combinatorial identity is proven by counting the number o
Law of large numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results
Structural induction
Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some other mathematical fields. It is a generalizati
Proof by infinite descent
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number,
Direct proof
In mathematics and logic, a direct proof is a way of showing thetruth or falsehood of a given statement by a straightforward combination ofestablished facts, usually axioms, existing lemmas and theore
Tombstone (typography)
In mathematics, the tombstone, halmos, end-of-proof, or Q.E.D. symbol "∎" (or "□") is a symbol used to denote the end of a proof, in place of the traditional abbreviation "Q.E.D." for the Latin phrase
Bijective proof
In combinatorics, bijective proof is a proof technique for proving that two sets have equally many elements, or that the sets in two combinatorial classes have equal size, by finding a bijective funct
Mathematical proof
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established
Proof of impossibility
In mathematics, a proof of impossibility is a proof that demonstrates that a particular problem cannot be solved as described in the claim, or that a particular set of problems cannot be solved in gen
Proof by exhaustion
Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split
Commutative diagram
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. It is said
List of long mathematical proofs
This is a list of unusually long mathematical proofs. Such proofs often use computational proof methods and may be considered non-surveyable. As of 2011, the longest mathematical proof, measured by nu
Proof by contradiction
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 contrad
Proof by intimidation
Proof by intimidation (or argumentum verbosum) is a jocular phrase used mainly in mathematics to refer to a specific form of hand-waving, whereby one attempts to advance an argument by marking it as o
Probabilistically checkable proof
In computational complexity theory, a probabilistically checkable proof (PCP) is a type of proof that can be checked by a randomized algorithm using a bounded amount of randomness and reading a bounde
Proof without words
In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying expl
Back-and-forth method
In mathematical logic, especially set theory and model theory, the back-and-forth method is a method for showing isomorphism between countably infinite structures satisfying specified conditions. In p
Of the form
In mathematics, the phrase "of the form" indicates that a mathematical object, or (more frequently) a collection of objects, follows a certain pattern of expression. It is frequently used to reduce th
Proofs from THE BOOK
Proofs from THE BOOK is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler. The book is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps
Constructive proof
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a n
Proof by contrapositive
In logic, the contrapositive of a conditional statement is formed by negating both terms and reversing the direction of inference. More specifically, the contrapositive of the statement "if A, then B"
Non-surveyable proof
In the philosophy of mathematics, a non-surveyable proof is a mathematical proof that is considered infeasible for a human mathematician to verify and so of controversial validity. The term was coined
Elementary proof
In mathematics, an elementary proof is a mathematical proof that only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysi
Wiles's proof of Fermat's Last Theorem
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a pr
Conditional proof
A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent.
Turing's proof
Turing's proof is a proof by Alan Turing, first published in January 1937 with the title "On Computable Numbers, with an Application to the Entscheidungsproblem". It was the second proof (after Church
Proofs That Really Count
Proofs That Really Count: the Art of Combinatorial Proof is an undergraduate-level mathematics book on combinatorial proofs of mathematical identies. That is, it concerns equations between two integer
Rigour
Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours
Probabilistic method
The probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object. It works by sh
Q.E.D.
Q.E.D. or QED is an initialism of the Latin phrase quod erat demonstrandum, meaning "which was to be demonstrated". Literally it states "what was to be shown". Traditionally, the abbreviation is place
Cantor's diagonal argument
In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, w
Original proof of Gödel's completeness theorem
The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 (and a shorter version of the proof, published as an article in 1930, titled "The completeness of the