Theorem

Theorem . It is a statement that must be demonstrated within a formal system. Proving theorems is a central issue in deductive mathematics . However, how the theorems arise is the task of intuitive mathematics [1]

Summary

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  • 1 Terminology
  • 2 Definitions of theorem
  • 3 Theorems in mathematical logic
  • 4 Theorems in other sciences
  • 5 Examples of theorems
  • 6 Utterances
    • 1 Usual form
    • 2 Implicative form
    • 3 Disjunctive form
  • 7 Sources
  • 8 References

Terminology

From the Latin word theorēma , a theorem is a proposition that must be proved logically from its hypothesis, resorting to axioms or other theorems previously demonstrated. This demonstration process is done using certain rules of inference.

The theorem is, therefore, a statement of importance. There are lower-level statements, such as the lemma (a statement linked to a longer theorem and guides the proof of this), the corollary ‘ (the statement that immediately follows the theorem) or the proposition (a result that is not found associated with no specific theorem); finally, scholium which is an important propositional observation.

It should be noted that, until the statement is proven, it is called either an unproven conjecture or proposition.

A theorem generally has a number of premises that must be listed or clarified beforehand. Then there is a conclusion, a mathematical statement, which is true under the given conditions. The informative content of the theorem is the relationship between the hypothesis and the thesis or conclusion.

Corollary . A logical statement that is an immediate consequence of a theorem, and can be proved using the properties of the theorem previously demonstrated.

Definitions of theorem

  • A theorem is a statement that can be proved to be true within a logical framework.
  • Theorem is a Propositionthat needs to be proved to be evident. For example: The sum of the angles of a triangle equals two right angles.
  • Proposition that affirms a demonstrable truth.
  • For every set, there is a set that does not belong to it. Proof. Let A be any set. Let D be the set of y that belong to set A, such that they fulfill the property “and does not belong to y”. …
  • It is a truth not evident, but demonstrable.

Theorems in mathematical logic

A theorem requires a logical framework; This framework will consist of a set of Axioms Axiomatic System and an Inference process, which allows to derive theorems from the axioms and theorems that have been previously derived.

In Mathematical Logic and Propositional Logic , any proved statement is called a theorem. More specifically in mathematical logic a finite sequence of well-formed Formulas (well-formed logical formulas) F1, …, Fn is called proof , such that each Fi is either an axiom or a theorem that follows from two previous formulas Fj and Fk (such that j <i and k <i) using a deduction rule.

Given a proof like the one above if the final element Fn is not an axiom then it is a theorem. Summing up, the above can be formally said, a theorem is a well-formed formula, which is not an axiom, and which may be the final element of some proof, that is, a theorem is a well-formed logical formula for which there is a proof .

Theorems in other sciences

Frequently in Physics or Economics some important statements that can be deduced or justified from other basic statements or hypotheses are commonly called theorems. However, frequently the areas of knowledge where these statements appear have often not been properly formalized in the form of an axiomatic system, so the term theorem should strictly be used with caution to refer to these demonstrable or deductible statements of “more basic” assumptions.

Examples of theorems

Dual theorem. The principle of duality affirms that from any theorem or construction of projective geometry we can obtain another, known as the Dual Theorem, only the words point and line can be interchanged, also modifying the relationships between the points and the lines. So, by this principle:

  • A point becomes a line.
  • Aligned points become lines that pass through a point.
  • Tangent lines become the point of tangency.
  • A circumscribed circle becomes an inscribed circle.

The dual theorem the theorem Pascal is the theorem Brianchon , Theorem Feuerbach . The Circumference Euler or 9 points, is tangent to the inscribed circumferences and exinscrita to the triangle.

  • Gauss theorem. The midpoints of the diagonals of an entire quadrilateral are in a straight line.
  • Euler’s theorem. In any convex polyhedron , the number of faces plus the number of vertices is equal to the number of edges plus two. (faces + vertices = edges + 2).

One of the best known theorems is Thales’ Theorem , which indicates that, by drawing a Line parallel to any of its sides in a triangle, two similar triangles are obtained (that is, they have equal angles and their sides are proportional).

Another very popular theorem is the Pythagorean Theorem , which states that, in a right triangle, the square of the Hypotenuse (the side with the longest length and opposite to the right angle), is equal to the sum of the squares of the legs ( the two minor sides of the right triangle).

Statements

Usual way

when the hypothesis and the thesis are not properly distinguished;

as an example “an exterior angle of a triangle is equal to the sum of the nonadjacent triangle angles” [2]

Implicative form

they are stated in the form if p , then q , being p the hypothesis and q , the conclusion or thesis.

as an example: if the natural number p is a multiple of 8, then p is a multiple of 4

Disjunctive form

is stated as p or not q , p being the hypothesis and q the conclusion.

For example: the triangle ACB is a rectangle with the right angle at C or the sum of the squares of the legs is not equal to the square of the hypotenuse.

 

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