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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.
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The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.



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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.


A proven mathematical statement.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.


Maths Logic a statement or formula that can be deduced from the axioms of a formal system by means of its rules of inference
Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005
References in periodicals archive ?
Although Theorem 1.1 may look somewhat artificial, it plays an important role in [12], [13], and [11].
Placing squares [x.sup.2], [y.sup.2] and [z.sup.2] on the sides of the triangle (Figure 2b) gives the 'windmill' representation of the two-dimensional Pythagorean theorem for [(x + [DELTA]).sup.2] + [(y + [DELTA]).sup.2] = [z.sup.2] (Figure 7b).
Recently, Dalal and Govil [15] unified the above theorems and proved the following theorem.
In a similar manner, in proof of Theorem 11, we obtain the result (47).
However, the condition on L(X) to be a barrelled space is very restrictive: the space L(X) is barrelled if and only if X is discrete; see Theorem 6.4 of [4].
For example, if [[chi].sub.1] and [[chi].sub.2] are non-equivalent characters, and the numbers [[alpha].sub.1] and [[alpha].sub.2] are algebraically independent over Q, then the function L(s, [[chi].sub.1]) L(s, [[chi].sub.2]) [zeta](s, [[alpha].sub.1]) [zeta](s, [[alpha].sub.2]) satisfies the hypotheses of Theorem 8, since a polynomial p(s) has a preimage (1,1,1,p(s)) [member of] [S.sup.2] x [H.sup.2](D), and the algebraic independence of the numbers [[alpha].sub.1] and [[alpha].sub.2] implies the linear independence of the set L(P, [[alpha].sub.1], [[alpha].sub.2]).
In Section 4, we give a similar estimate in the case that H is a p-torus subgroup of a particular group G and as application, we prove a topological Tverberg type theorem for any natural number, which is a weak version of the famous topological Tverberg conjecture.
In Section 3, we use the time scales Taylor formula, see [11, Theorem 1.113], i.e.,
From formula ([I.sub.4]) in Theorem 4, for [mathematical expression not reproducible], we get
Observe that if in Theorem 16 we have [alpha] = 1, the statement of Theorem 16 becomes the statement of Theorem 10.
Emami, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal., 72 (2010), 2238-2242.