Theorem

See also: Theorem (homonymy)

A theorem is a proposal which can be mathematically shown, i.e. a Assertion which can be established as true through a reasoning Logique builds starting from Axiome S. a theorem is to be distinguished from a Théorie.

Once the shown theorem, it is regarded as truth whatever the value of truth of its premise (basic assumption) because it is appeared as an implication: if has then B is true is necessarily true. It can then be used to show other proposals. To show the theorem consists in at the same time showing impossibility of having has true and B false.

A theorem generally has:

• of the basic assumptions, i.e. of the conditions which can be enumerated in the theorem or described in advance,
• a conclusion, i.e. a mathematical assertion which is true under the basic conditions.

The demonstration, although necessary to the classification of the proposal like “theorem”, is not regarded as belonging to the theorem.

Another possible definition of a theorem : “a statement of which one can show exactitude. ”

The demonstration includes/understands:

• of the Axiom S or the Postulate S;
• the Assumption S of the theorem;
• of other already shown theorems.

Each stage of the proof is related to the preceding ones by logical rules of inference.

Examples of demonstrations

Irrationality of the square root of 2

A reductio ad absurdum regarded as one of most beautiful by Paul Erdös is the demonstration of the irrationality of $\backslash \; sqrt\; \{2\}$.

By the absurdity, thus let us suppose that $\backslash \; sqrt\; \{2\}$ is a rational . There exist two whole p and Q (strictly positive) such as $\backslash \; sqrt\; \{2\}\; =\; \backslash \; frac\; p\; q$.

Even if it means to simplify by PGCD of p and Q , one can suppose p and Q first between them (the Fraction $p/q$ is known as irreducible).

By raising squared the two members, one obtains:

$2\; =\; \backslash \; frac\; \{p^2\}\; \{q^2\}$

By multiplying by Q ² the two sides, one finds then:

$\backslash \; 2\; \backslash \; cdot\; q^2\; =p^2$

One from of deduced that 2 divides p ²= p × p and according to the Lemme de Gauss, since 2 is first, it results from it that 2 divides p , therefore it exists K an entirety such as p=2k . One finds then while simplifying by 2:

$\backslash \; q^2\; =2\; \backslash cdot\; k^2$

This equality shows, according to the lemma of Gauss, that 2 divides Q .

One has thus shown that 2 divides p and Q, which is contradictory with the starting assumption, where one had supposed p and Q first between them. CQFD.

" Fausse" reductio ad absurdum???

The demonstration presented above can " easily; retourner" in a simple and direct demonstration owing to the fact that $\backslash \; sqrt\; \{2\}$ is not rational (without lemma of Gauss, nor pgcd, etc.) ! Here an example of drafting. Let us consider two nonnull entireties $p,\; q$. Either $a$ the greatest entirety such as $2^a$ divides $p^2$. Then $a$ is even: indeed, $p\; =\; 2^k\; p\text{'}$ with $p\text{'}$ odd, therefore $p^2\; =\; 2^\; \{2k\}\; p\text{'}^2$ where $p\text{'}^2$ is odd, from where $a=2k$.

In the same way, the greatest entirety $b$ such as $2^b$ divides $2q^2$ is odd: indeed, $q\; =\; 2^n\; q\text{'}$ with $q\text{'}$ odd, therefore $2q^2\; =\; 2^\; \{1+2n\}\; q\text{'}^2$ where $q\text{'}^2$ is odd, from where $b=1+2n$. An even number (here $a$) is never equal to an odd number (here $b$), this is why one has $p^2\; \backslash \; neq\; 2\; q^2$. One finishes by $\backslash \; frac\; \{p^2\}\; \{q^2\}\; \backslash \; neq\; 2$, then $\backslash \; frac\; p\; Q\; \backslash \; neq\; \backslash \; pm\; \backslash \; sqrt\; \{2\}$ for any couple of entireties $p,\; q$ nonnull. From where the irrationality of $\backslash \; sqrt\; \{2\}$.

Theorems of geometry

In its work “Grundlagen der Geometrie” David Hilbert gives a new form to the Géométrie and in installation its bases.

Let us point out some of the axioms of the bases of the geometry:

• I, 3 On a right , there are at least two points; there exist at least three not aligned points.

• II, 2 Two points has and C being given, there exists at least a point B pertaining to the right-hand side AC and such as B is between has and C.
• II, 3 Of three points of a line, it does not have of them there more one which is between the two others.
• II, 4 Have, B and C three not aligned points and have a line of the ABC plan which does not pass by any the points has, B and C; if the line has master key by one of the points of segment AB, then it passes or by a point of segment BC or a point of the segment AC.

Let us show the second theorem:

; Theorem

Two points has and C being given, there exists on the line (AC) at least a point D located between has and C (i.e. on).

; Demonstration

Let us consider the line AC, according to axiom I, 3, it exists at least a point E external on this line AC. According to axiom II, 2, on line AE there exists at least a point F such as E either ranging between has and F in other words such as E or a point of segment AF. According to the same axiom, on line FC, there exists at least a point G such as C is on segment FG. According to II, 3, the point G is thus external with segment FC (if not C and G are two points located between F etG). According to axiom II, 4 line EG cuts inevitably the segment AC in a point D. c.q.f.d.

Other forms of assertions

In the broad sense any actually shown assertion can take the name of theorem. In the works of mathematics, it is however of use to hold this term with the assertions considered as particularly interesting or important. According to their importance or their utility, the other assertions can take different names:

• lemma: assertion being used as intermediary to show a more important theorem;
• corollary: result which rises directly from a proven theorem;
• proposal: relatively simple result which is not associated with a particular theorem;
• note: result interesting or consequence which can belong to the proof or another assertion;
• Conjecture: mathematical proposal which one is unaware of the value of truth. Once proven, a conjecture becomes a theorem.

Like statement above, a theorem requires a logical reasoning based on axioms. That consists of a series of fundamental axioms (see Système of axioms) and a process of inference which makes it possible to derive the axioms in new theorems and other theorems shown before. In the Logical of the proposals, any shown assertion is called a theorem.

See too

• List of the theorems for a list of famous theorems and Conjecture S

Be-X-old: Тэарэма Simple: Theorem

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