Proofs Proofs In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. The assumed axioms are ZFC unless indicated otherwise. ...more on Wikipedia about "Mathematical proof"
In mathematics, the method of considering a minimal counterexample combines the ideas of inductive proof and proof by contradiction. Abstractly, in trying to prove a proposition P, one assumes that it is false, and that therefore there is at least one counterexample. With respect to some idea of size, which may need to be chosen skilfully, one assumes that there is such a counterexample C that is minimal. We expect that C is something quite hypothetical (since we are trying to prove P), but it may be possible to argue that if C existed, it would have some definite properties. From those we then try to get a contradiction. ...more on Wikipedia about "Minimal counterexample"
In mathematics, a nonconstructive proof, as opposed to a constructive proof, is a mathematical proof that purports to demonstrate the existence of something, but does not reveal how to construct it. ...more on Wikipedia about "Nonconstructive proof"
To see the equivalence, note first that if Theorem 1 holds, and φ is not satisfiable in any structure, then ¬φ is valid in all structures and therefore provable, thus φ is refutable and Theorem 2 holds. If on the other hand Theorem 2 holds and φ is valid in all structures, then ¬φ is not satisfiable in any structure and therefore refutable; then ¬¬φ is provable and then so is φ, thus Theorem 1 holds. ...more on Wikipedia about "Original proof of Gödel's completeness theorem"
Given a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of interior points located in the polygon and the number b of boundary points placed on the polygon's perimeter: ...more on Wikipedia about "Pick's theorem"
:This article is not about probabilistic algorithms, which give the right answer with high probability but not with certainty, nor about Monte Carlo methods, which are simulations relying on pseudo-randomness. ...more on Wikipedia about "Probabilistic method"
Proof by exhaustion, also known as the brute force method or case analysis, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases, and each case is proved separately. A proof by exhaustion contains two stages: ...more on Wikipedia about "Proof by exhaustion"
Proof by tenure is a derogatory description for a flawed mathematical proof presented in class by a professor. The term does not apply to simple mistakes such as poor arithmetic, but instead tends to refer to a proof in which the professor makes a more fundamental mistake, e.g. relying on a logical fallacy. If the proof is questioned during or after class, the professor may not immediately acknowledge the flaw. ...more on Wikipedia about "Proof by tenure"
Proof by verbosity is a term used to describe an excessively verbose mathematical proof that may or may not actually prove the result. Such proofs are most often presented by students who don't fully grasp the concepts they are writing about. Students presenting such proofs often either hope to hide their lack of understanding or are uncertain how extensive their proof is expected to be. ...more on Wikipedia about "Proof by verbosity"
(Proof of Bertrand's postulate) Lemma ...more on Wikipedia about "Proof of Bertrand's postulate"
A simplified version is given here. This proof does not use the standard mathematical symbols for there exists and for all to make it more accessible to less mathematically motivated readers. The key technique is natural deduction logic and proof by contradiction. ...more on Wikipedia about "Proof of mathematical induction"
This article presents background and proofs of the fact that the recurring decimal 0.9999… equals 1, not approximately but exactly. ...more on Wikipedia about "Proof that 0.999... equals 1"
The rational number 22/7 is a widely used approximation of π. It is a convergent in the simple continued fraction expansion of π. It is greater than π, as can be readily seen in the decimal expansion of these values: ...more on Wikipedia about "Proof that 22 over 7 exceeds π"
In mathematics, the series expansion of the number e ...more on Wikipedia about "Proof that e is irrational" Made by www.shortopedia.com.
Let us think of each such string as representing a bracelet. That is, we connect the two ends of the string together, and regard two strings as the same bracelet if we can rotate one string to obtain the second string; in this case we will say that the two strings are friends. In our example, the following strings are all friends: ...more on Wikipedia about "Proofs of Fermat's little theorem"
The Pythagorean trigonometric identity is a trigonometric identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae (see trigonometric identity#Angle sum and difference identities) it is the basic relation among the sin and cos functions from which all others may be derived (see trigonometric function#Other definitions for the relevant theorem). ...more on Wikipedia about "Pythagorean trigonometric identity"
Q.E.D. is an abbreviation of the Latin phrase "quod erat demonstrandum" (literally, "which was to be demonstrated"). This is a translation of the Greek (hóper édei deĩxai) which was used by many early mathematicians including Euclid and Archimedes. Q.E.D. may be written at the end of mathematical proofs to show that the result required for the proof to be complete has been obtained. It is not seen as frequently now as in earlier centuries. ...more on Wikipedia about "Q.E.D."
In combinatorics, Ramsey's theorem states that in colouring a large complete graph (that is a simple graph, where an edge connects every pair of vertices), one will find complete subgraphs all of the same colour. In a precise statement, for any pair of positive integers (r,s), there exists an integer R(r,s) such that for any complete graph on R(r,s) vertices, whose edges are coloured red or blue, there exists either a complete subgraph on r vertices which is entirely blue, or a complete subgraph on s vertices which is entirely red. Here R(r,s) signifies an integer that depends on both r and s. ...more on Wikipedia about "Ramsey's theorem"
Reductio ad absurdum ( Latin for "reduction to the absurd", traceable back to the Greek ἡ εις άτοπον απαγωγη (hi eis átopon apagogi), "reduction to the impossible", often used by Aristotle), also known as an apagogical argument or reductio ad impossibile, is a type of logical argument where we assume a claim for the sake of argument, arrive at an absurd result, and then conclude the original assumption must have been wrong, since it gave us this absurd result. This is also known as proof by contradiction. It makes use of the law of non-contradiction — a statement cannot be both true and false. In some cases it may also make use of the law of excluded middle — a statement which cannot be false, must then be true. ...more on Wikipedia about "Reductio ad absurdum"
Input: N, the integer to be factored, which must be neither a prime number nor a perfect square. ...more on Wikipedia about "Shanks' square forms factorization"
In combinatorial mathematics, Sperner's lemma states that every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors. The initial result of this kind was proved by Emanuel Sperner, in relation with proofs of invariance of domain. Sperner colorings have been used for effective computation of fixed points, in root-finding algorithms, and are applied in fair division algorithms. ...more on Wikipedia about "Sperner's lemma" Please visit again http://www.shortopedia.com
Structural induction is a proof method that is used in mathematical logic (e.g., the proof of Los's theorem), computer science, graph theory, and some other mathematical fields. It is a generalization of mathematical induction. Structural recursion is a recursion method bearing the same relationship to structural induction as ordinary recursion bears to ordinary mathematical induction. ...more on Wikipedia about "Structural induction"
In geometry, Thales' theorem (named after Thales of Miletus) states that if A, B and C are points on a circle where the line AC is a diameter of ...more on Wikipedia about "Thales' theorem"
The friendship theorem is a mathematical theorem in an area of mathematics called Ramsey theory. ...more on Wikipedia about "Theorem on friends and strangers"
# The basis for induction is trivial; the substantial part of the proof goes from case n to case n + 1. ...more on Wikipedia about "Three forms of mathematical induction"
This article is licensed under the GNU Free Documentation License.
It uses material from the Wikipedia . Direct links to the original articles are in the text.
If you use exact copy or modified of this article you should preserve above paragraph and put also : It uses material from
the Shortopedia article about "Proofs".
| MAIN PAGE | MAIN INDEX | CONTACT US |