Proofs Proofs Automated theorem proving (currently the most important subfield of automated reasoning) is the proving of mathematical theorems by a computer program. Depending on the underlying logic, the problem of deciding the validity of a theorem varies from trivial to impossible. For the frequent case of propositional logic, the problem is decidable but NP-complete, and hence only exponential-time algorithms exist for general solutions. For first-order logic it is recursively enumerable, i.e., given unbounded resources, any valid theorem can eventually be proven. Invalid statements, i.e. formulas that are not entailed by a given theory, cannot always be recognized. In these cases, a first-order theorem prover will fail to terminate while searching for a proof. Despite these theoretical limits, practical theorem provers can solve many hard problems in these logics. ...more on Wikipedia about "Automated theorem proving"
Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. (It is also called the diagonalization argument or the diagonal slash argument or the diagonal method.) ...more on Wikipedia about "Cantor's diagonal argument"
In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group on G. This can be understood as an example of the group action of G on the elements of G. ...more on Wikipedia about "Cayley's theorem"
In mathematics, the exponential function can be characterized in many ways. The following three characterizations (definitions) are most common. This article discusses why each characterization makes sense, and why the characterizations are independent of and equivalent to each other. As a special case of these considerations, we will see that the three most common definitions given for the E (mathematical constant) are also equivalent to each other. ...more on Wikipedia about "Characterizations of the exponential function"
In mathematics, an equation or system of equations is said to have a closed-form solution if, and only if, at least one solution can be expressed analytically in terms of a bounded number of well-known operations. The classic example involves the two roots of a quadratic equation, which can be expressed in closed form in terms of addition and subtraction, multiplication and division, and square root extraction. ...more on Wikipedia about "Closed-form expression"
A combinatorial proof is a method of proving a statement, usually a combinatorics identity, by counting some carefully chosen object in different ways to obtain different expressions in the statement (see also double counting). Since those expressions count the same object, they must be equal to each other and thus the statement is established. ...more on Wikipedia about "Combinatorial proof"
Strong induction, also known as complete induction, is a variant on the principle of mathematical induction. The inductive hypothesis, instead of being simply ...more on Wikipedia about "Complete induction"
A computer-assisted proof is a mathematical proof that has been generated by computer. ...more on Wikipedia about "Computer-assisted proof"
A conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditional necessarily leads to the conclusion. Proving this requires assuming the premise and deriving, from that assumption, the consequent of the conditional. By proving the connection between the antecedent and the consequent, the assumption of the antecedent is justified post hoc. ...more on Wikipedia about "Conditional proof"
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object. This is in contrast to a nonconstructive proof (also known as an existence proof or pure existence theorem) which proves the existence of a mathematical object with certain properties, but does not provide a means of constructing an example. Constructivism is the philosophy that rejects all but constructive proofs in mathematics. ...more on Wikipedia about "Constructive proof"
(Degree of anonymity) ; Internal/External : an internal attacker controls nodes in the network, whereas an external can only compromise communication channels between nodes. ...more on Wikipedia about "Degree of anonymity"
Diagram chasing is a method of mathematical proof used especially in homological algebra. Given a commutative diagram, a proof by diagram chasing involves formally using the properties of the diagram, such as injective or surjective maps, or exact sequences. One ends up "chasing" elements around the diagram, until the desired element is constructed. ...more on Wikipedia about "Diagram chasing"
In mathematics, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually existing lemmas and theorems, without making any further assumptions. ...more on Wikipedia about "Direct proof"
In combinatorial mathematics, the Erdős–Ko–Rado theorem of Paul Erdős, Chao Ko, and Richard Rado is the following. If , and is a family of subsets of , such that each subset is of size , and each pair of subsets intersects, then the maximum number of sets that can be in is given by the binomial coefficient ...more on Wikipedia about "Erdős–Ko–Rado theorem"
In mathematics, an existence theorem is a theorem with a statement beginning 'there exist(s) ..', or more generally 'for all x, y, ... there exist(s) ...'. That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. Many such theorems will not do so explicitly, as usually stated in standard mathematical language. For example, the statement that the sine function is continuous; or any theorem written in big O notation. The quantification is then hidden in definitions. ...more on Wikipedia about "Existence theorem"
In Riemannian geometry, the fundamental theorem of Riemannian geometry states that given a Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free connection preserving the metric tensor. ...more on Wikipedia about "Fundamental theorem of Riemannian geometry"
In geometry, Helly's theorem is a basic combinatorial result on convex sets. It was proved by Eduard Helly, and gave rise to the notion of Helly family. ...more on Wikipedia about "Helly's theorem"
In geometry, Heron's formula (also called Hero's formula) states that the area of a triangle whose sides have lengths a, b and c is ...more on Wikipedia about "Heron's formula"
(Holomorphic functions are analytic) In complex analysis, a complex-valued function f of a complex variable ...more on Wikipedia about "Holomorphic functions are analytic"
In mathematics, a proof by infinite descent is a particular kind of proof by mathematical induction. One typical application is to show that a given equation has no solutions. Assuming a solution exists, one shows that another exists, that is in some sense 'smaller'. Then one must show, usually with greater ease, that the infinite descent implied by having a whole sequence of solutions that are ever smaller, by our chosen measure, is an impossibility. This is a contradiction, so no such initial solution can exist. ...more on Wikipedia about "Infinite descent"
Most of these proofs depend on some variation of the same error. The error is to take a function f that is not one-to-one, to observe that f(x) = f(y) for some x and y, and to (erroneously) conclude that therefore x = y. Division by zero is a special case of this; the function f is x → x × 0, and the erroneous step is to start with x × 0 = y × 0 and to conclude that therefore x = y. ...more on Wikipedia about "Invalid proof"
König's lemma or König's infinity lemma is a theorem in graph theory due to Denes König that states the following. ...more on Wikipedia about "König's lemma"
In mathematics, Leibniz' formula for π, due to Gottfried Leibniz, states that ...more on Wikipedia about "Leibniz formula for pi"
* Transcendence of e and π (as corollaries of Lindemann-Weierstrass) ...more on Wikipedia about "List of mathematical proofs"
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence. A somewhat more general form of argument used in mathematical logic and computer science shows that expressions that can be evaluated are equivalent; this is known as structural induction. ...more on Wikipedia about "Mathematical induction" Things Go Better with http://www.shortopedia.com. Proofs
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