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

Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005

theorem

[′thir·əm]

(mathematics)

A proven mathematical statement.

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 It might be outdated or ideologically biased.

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.

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References in periodicals archive

Although Theorem 1.1 may look somewhat artificial, it plays an important role in [12], [13], and [11].

On normalization of quasi-log canonical pairs

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).

The Pythagorean geometric gear

Recently, Dalal and Govil [15] unified the above theorems and proved the following theorem.

Annular Bounds for the Zeros of a Polynomial

In a similar manner, in proof of Theorem 11, we obtain the result (47).

Generalized Fractional Integral Operators Involving Mittag-Leffler Function

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].

Free Subspaces of Free Locally Convex Spaces

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]).

On Zeros of Some Combinations of Dirichlet L-Functions and Hurwitz Zeta-Functions

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.

(H,G)-coincidence theorems for manifolds and a topological Tverberg type theorem for any natural number r

In Section 3, we use the time scales Taylor formula, see [11, Theorem 1.113], i.e.,

GENERAL HIGHER-ORDER DYNAMIC OPIAL INEQUALITIES WITH APPLICATIONS

From formula ([I.sub.4]) in Theorem 4, for [mathematical expression not reproducible], we get

Conditional Stability and Asymptotic Behavior of Solutions of Weakly Delayed Linear Discrete Systems in [R.sup.2]

Using Lemma 1.2 in [10] we can prove the following theorem.

A Conditional Fourier-Feynman Transform and Conditional Convolution Product with Change of Scales on a Function Space II

Observe that if in Theorem 16 we have [alpha] = 1, the statement of Theorem 16 becomes the statement of Theorem 10.

Hermite-Hadamard Type Inequalities Obtained via Fractional Integral for Differentiable m-Convex and ([alpha], m)-Convex Functions

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.

EXISTENCE OF COUPLED FIXED POINTS USING GENERALIZED ( , AY) -ORDER CONTRACTIVE MAPPINGS