theorem


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theorem,

in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiomaxiom,
in mathematics and logic, general statement accepted without proof as the basis for logically deducing other statements (theorems). Examples of axioms used widely in mathematics are those related to equality (e.g.
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 in that a proofproof,
in mathematics, finite sequence of propositions each of which is either an axiom or follows from preceding propositions by one of the rules of logical inference (see symbolic logic).
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 is required for its acceptance. A lemma is a theorem that is demonstrated as an intermediate step in the proof of another, more basic theorem. A corollary is a theorem that follows as a direct consequence of another theorem or an axiom. There are many famous theorems in mathematics, often known by the name of their discoverer, e.g., the Pythagorean Theorem, concerning right triangles. One of the most famous problems of number theory was the proof of Fermat's Last Theorem (see Fermat, Pierre deFermat, Pierre de
, 1601–65, French mathematician. A magistrate whose avocation was mathematics, Fermat is known as a founder of modern number theory and probability theory.
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); the theorem states that for an integer n greater than 2 the equation xn+yn=zn admits no solutions where x, y, and z are also integers.

Theorem

 

a statement, in some deductive theory, that has been or is to be proved (seeDEDUCTION). Examples of deductive theories are provided by mathematics, logic, theoretical mechanics, and some branches of physics. Every such theory consists of theorems that are proved one after another on the basis of previously proved theorems. The first statements in the deductive process are accepted without proof and thus form the logical basis of the given area of the theory. Such primitive statements are called axioms.

In the formulation of a theorem, a distinction is made between the hypothesis and the conclusion. Consider, for example, the following two theorems: (1) If the sum of the digits in a number is divisible by 3, then the number is itself divisible by 3. (2) If one of the angles in a triangle is a right angle, then the other two angles are acute. In these examples the word “if” is followed by the hypothesis of the theorem, and the word “then” is followed by the conclusion. Every theorem can be expressed in this form. For example, the theorem “Any angle inscribed in a semicircle is a right angle” can be expressed “If an angle is inscribed in a semicircle, then the angle is a right angle.”

The converse of a theorem expressed in the form “If..., then...” is obtained by interchanging the hypothesis and the conclusion. A theorem and its converse are converses of each other. In general, the validity of a theorem does not imply the validity of its converse. For example, the converse of theorem (1) is true, but the converse of theorem (2) is false. If a theorem and its converse are both true, then the hypothesis of either theorem is a necessary and sufficient condition for the validity of the conclusion (seeNECESSARY AND SUFFICIENT CONDITIONS).

If the hypothesis and conclusion of a theorem are replaced by their negations, then the inverse of the given theorem is obtained. The inverse of a theorem is equivalent to the theorem’s converse. Moreover, the converse of the inverse of a theorem is equivalent to the original theorem. Consequently, the validity of a theorem can be demonstrated by both a direct and an indirect proof. An indirect proof, also known as reductio ad absurdum, involves showing that the negation of the hypothesis of the theorem follows from the negation of the theorem’s conclusion. This method of proof is very widely used in mathematics.

theorem

[′thir·əm]
(mathematics)
A proven mathematical statement.

theorem

Maths Logic a statement or formula that can be deduced from the axioms of a formal system by means of its rules of inference
References in periodicals archive ?
dagger]] has rational singularities by [FT12, Theorem 6.
Observe that if in Theorem 14 we have [alpha] = 1, the statement of Theorem 14 becomes the statement of Theorem 8.
Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations.
Theorem 2 [9] A connected 4-regular graph G has a [P.
Theorem's business strategy is to offer our clients the very best available services in every research area," said John Potthoff, Theorem president and chief executive officer.
For example, a very simple proof of this theorem was given in reference [5] recently.
Theorem has been solving data compatibility issues for over 20 years for many of the leading automotive, aerospace, defence and power generation companies who design and share data using a variety of data types.
Pati (2007) established two Tauberian theorems which are more general than a theorem of Pati (2002) and a theorem of Littlewood (1967).
If you are a Theorem Solutions JT - CATIA V5 maintained customer you will be able to download the free upgrade to your Theorem Solutions CADverter ensuring that it remains compatible with the latest Ford approved software and process.
Rumely's introduction to this exposition on the Fekete-Szego theorem includes a sketch of the basic proof and contextualizes its development with the Robinson theorem on totally real algebraic integers in an interval and Cantor's extension with splitting conditions.
s] x f), the next result is not immediately contained in the right module version of Theorem 2.