Theorem proving

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Proof by contradiction

A proof by contradiction establishes the truth of a given proposition by the supposition that it is false and the subsequent drawing of a conclusion that is contradictory to something that is proven to be true. That is, the supposition that is false followed necessarily by the conclusion from not-, where is false, which implies that is true.For example, the second of Euclid's theorems starts with the assumption that there is a finite number of primes. Cusik gives some other nice examples.

Uniqueness theorem

A theorem, also called a unicity theorem, stating the uniqueness of a mathematical object, which usually means that there is only one object fulfilling given properties, or that all objects of a given class are equivalent (i.e., they can be represented by the same model). This is often expressed by saying that the object is uniquely determined by a certain set of data. The word unique is sometimes replaced by essentially unique, whenever one wants to stress that the uniqueness is only referred to the underlying structure, whereas the form may vary in all ways that do not affect the mathematical content.The object of many uniqueness theorems is the solution to a problem or an equation; in such cases, a uniqueness theorem is normally combined with an existence theorem.

Existence theorem

A theorem stating the existence of an object, such as the solution to a problem or equation. Strictly speaking, it need not tell how many such objects there are, nor give hints on how to find them. Some existence theorems give explicit formulas for solutions (e.g., Cramer's rule), others describe in their proofs iteration processes for approaching them (e.g., Bolzano-Weierstrass theorem), while others are settled by nonconstructive proofs which simply deduce the necessity of solutions without indicating any method for determining them (e.g., the Brouwer fixed point theorem, which is proved by reductio ad absurdum, showing that the nonexistence would lead to a contradiction).

Transfinite induction

Transfinite induction, like regular induction, is used to show a property holds for all numbers . The essential difference is that regular induction is restricted to the natural numbers , which are precisely the finite ordinal numbers. The normal inductive step of deriving from can fail due to limit ordinals.Let be a well ordered set and let be a proposition with domain . A proof by transfinite induction uses the following steps (Gleason 1991, Hajnal 1999): 1. Demonstrate is true. 2. Assume is true for all . 3. Prove , using the assumption in (2). 4. Then is true for all . To prove various results in point-set topology, Cantor developed the first transfinite induction methods in the 1880s. Zermelo (1904) extended Cantor's method with a "proof that every set can be well-ordered," which became the axiom of choice or Zorn's Lemma (Johnstone 1987). Transfinite induction and Zorn's lemma are often used interchangeably (Reid 1995), or are strongly..

Principle of mathematical induction

The truth of an infinite sequence of propositions for , ..., is established if (1) is true, and (2) implies for all . This principle is sometimes also known as the method of induction.

Pattern of two loci

According to G. Pólya, the method of finding geometric objects by intersection. 1. For example, the centers of all circles tangent to a straight line at a given point lie on a line that passes through and is perpendicular to . 2. In addition, the circle centered at with radius is the locus of the centers of all circles of radius passing through . The intersection of and consists of two points and which are the centers of two circles of radius tangent to at .Many constructions with straightedge and compass are based on this method, as, for example, the construction of the center of a given circle by means of the perpendicular bisector theorem.

Modus tollens

Modus tollens is a valid argument form in propositional calculus in which and are propositions. If implies , and is false, then is false. Also known as an indirect proof or a proof by contrapositive.For example, if being the king implies having a crown, not having a crown implies not being the king.

Characterization

A description of an object by properties that are different from those mentioned in its definition, but are equivalent to them. The following list gives a number of examples.1. A rational number is defined as the quotient of two integers, but it can be characterized as a number admitting a finite or repeating decimal expansion. 2. An equilateral triangle is defined as a triangle having three equal sides, but it can be characterized as a triangle having two angles of . 3. A real square matrix is nonsingular, by definition, if it admits a matrix inverse, but it can be characterized by the condition that its determinant be nonzero. Of course, a characterization should not merely be a rephrasing of the definition, but should give an entirely new description, which is useful because it contains a simpler formulation, can be verified more easily, is interesting because it places the object in another context, or unveils unexpected links between different..

Deduction theorem

A metatheorem in mathematical logic also known under the name "conditional proof." It states that if the sentential formula can be derived from the set of sentential formulas , then the sentential formula can be derived from .In a less formal setting, this means that if a thesis can be proven under the hypotheses , then one can prove that implies under hypothesis .

Theorem

A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is an embodiment of some general principle that makes it part of a larger theory. The process of showing a theorem to be correct is called a proof.Although not absolutely standard, the Greeks distinguished between "problems" (roughly, the construction of various figures) and "theorems" (establishing the properties of said figures; Heath 1956, pp. 252, 262, and 264).According to the Nobel Prize-winning physicist Richard Feynman (1985), any theorem, no matter how difficult to prove in the first place, is viewed as "trivial" by mathematicians once it has been proven. Therefore, there are exactly two types of mathematical objects: trivial ones, and those which have not yet been proven.The late mathematician P. Erdős has often been associated with the observation..

Lemma

A short theorem used in proving a larger theorem. Related concepts are the axiom, porism, postulate, principle, and theorem.The late mathematician P. Erdős has often been associated with the observation that "a mathematician is a machine for converting coffee into theorems" (e.g., Hoffman 1998, p. 7). However, this characterization appears to be due to his friend, Alfred Rényi (MacTutor, Malkevitch). This thought was developed further by Erdős' friend and Hungarian mathematician Paul Turán, who suggested that weak coffee was suitable "only for lemmas" (MacTutor, Malkevitch).

Proof

A rigorous mathematical argument which unequivocally demonstrates the truth of a given proposition. A mathematical statement that has been proven is called a theorem.According to Hardy (1999, pp. 15-16), "all physicists, and a good many quite respectable mathematicians, are contemptuous about proof. I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite uninteresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for proof.... [This opinion], with which I am sure that almost all physicists agree at the bottom of their hearts, is one to which a mathematician ought to have some reply."To prove Hardy's assertion, Feynman is reported to have commented, "A great deal more is known than has been proved" (Derbyshire 2004, p. 291).There is some debate among mathematicians..

Q.e.d.

"Q.E.D." (sometimes written "QED") is an abbreviation for the Latin phrase "quod erat demonstrandum" ("that which was to be demonstrated"), a notation which is often placed at the end of a mathematical proof to indicate its completion. Several symbols are occasionally used as synonyms for Q.E.D. These include a filled square (Unicode U+220E, as used in Mathematics Magazine and American Mathematical Monthly), a filled rectangle (Knuth 1997, pp. 3 and 39), or an empty square .

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