Sentence (mathematics)

In mathematics, a sentence or theorem is a consistent logical statement that can be recognized as true by means of a proof , that is, can be derived from axioms , definitions and already known sentences.

A sentence is often referred to differently depending on its role, meaning or context. Within an article or a monograph (e.g. a dissertation or a textbook) one uses

Lemma (or auxiliary clause ) for a statement that is only used in the proof of other sentences in the same work and has no meaning regardless of it,
Proposition for a statement that is also mainly locally significant, such as a proposition that is used in more than one proof,
Theorem (or theorem ) for an essential knowledge presented in the work, and
Corollary (or sequential block ) for a trivial conclusion, the results from a set or definition without great effort.

The classification of a sentence in one of the above categories is subjective and has no consequences for the use of the sentence. Many authors dispense with the term proposition and use lemma or theorem for it. Also corollary is not always set distinguished. On the other hand, it is quite common and helpful for the reader if pure auxiliary clauses are recognizable as such.

Sentences that are generally known and are usually not cited with the original source bear the name of the subject matter about which they are making a statement, or the name of the author, or both. The terms fundamental clause or main clause (a field of mathematics) are also used in this context , and the distinction between proposition and lemma has often grown historically rather than determined by content and meaning. Many examples of such names can be found in the List of Mathematical Theorems .

Examples of sentences

Below are a few simple sentences. The calculus to be used is given in brackets.

If every human is mortal and Socrates is human, then Socrates is mortal. ( Predicate logic ).

Every non-empty set has at least one element . ( Set theory )

The sum of the interior angles of a triangle is 180 degrees . ( Euclidean geometry )

For every real number there is a larger natural number . ( Archimedean order , analysis )

There is no such thing as a rational number whose square is 2. ( Number theory )

Let it be steady . Then it is also steady. (Analysis) ${\ displaystyle f, g: \ mathbb {R} \ to \ mathbb {R}}$ ${\ displaystyle f \ circ g: \ mathbb {R} \ to \ mathbb {R}}$

construction

formulation

Although a mathematical set of a statement can be made of any shape (eg "not V or A ."), A mathematical theorem mostly in the subjunctive formulated requirement and as a declarative sentence worded statement divided (example: "Be V . Then A. "), So that the impression of an implication arises.

Caution: Inconsiderate removal and use of individual parts of a sentence can lead to false conclusions, as these parts generally do not have to be valid.

Examples

" " ${\ displaystyle n \ notin \ mathbb {N} \ quad \ vee \ quad n {\ mbox {is not prime}} \ quad \ vee \ quad n = 2 \ quad \ vee \ quad n {\ mbox {is odd} }}$
“Let n be a prime number . For n the following applies: " ${\ displaystyle n = 2 \ quad \ vee \ quad n \ in 2 \ cdot \ mathbb {N} +1}$
“ When it rains, the road gets wet. "(Not a sentence in the mathematical sense)
From the plane geometry: “ If a real square is a parallelogram , then opposite sides are the same length. ”(Here,“ real square ”means that degenerate and overturned squares are excluded from consideration).

Reverse theorem

If you swap the premise and statement of the sentence in a sentence , you get the corresponding reverse sentence. These are logical statements of the form “ requirement ⇐ statement ”. A distinction must then be made between the following cases:

If the reverse clause is not a clause - that is, it is false - then the premise of the clause is sufficient, but not necessary .
If the reverse sentence is a sentence - that is, it is applicable - then the premise of the sentence is necessary and sufficient. In this case one can formulate a further sentence in which the premise and statement of the sentence are equivalent (example: “ V holds, if and only if A holds ”).

Examples

“ If the street is wet, then it has rained. “This reverse sentence is wrong, because the water could have got onto the street differently. The prerequisite for the sentence “ it rained ” is therefore sufficient, but not necessary .
“ If opposite sides of a real square are the same length, then it's a parallelogram. “This reverse sentence is true. The premise of the sentence is necessary and sufficient . The sentence and the reverse sentence can be summarized: “ A real square is a parallelogram if and only if the opposite sides are the same length. "

Dependence on the division into prerequisite and statement

It is possible to have the same logical statement in various ways condition and statement divide, and the reversal rate depends on this division.

The logical statement can be written as a sentence in the following ways, for example: ${\ displaystyle \ lnot A \ lor \ lnot B \ lor C}$

${\ displaystyle (A \ land B) \ Rightarrow C}$ - reverse theorem: ${\ displaystyle C \ Rightarrow (A \ wedge B) \ quad \ equiv \ quad (A \ vee \ neg C) \ wedge (B \ vee \ neg C)}$
${\ displaystyle A \ Rightarrow (\ lnot B \ lor C)}$ - reverse theorem: ${\ displaystyle (\ lnot B \ lor C) \ Rightarrow A \ quad \ equiv \ quad (A \ vee B) \ wedge (A \ vee \ neg C)}$