Lemma (mathematics)

A lemma or lemma ( gr λῆμμα, taking ',' acceptance ', plural: " lemmas " ) is a mathematical or logical statement that is used in the proof of a theorem, which is not itself given the rank of a sentence. The distinction of propositions and lemmas is fluid and not objective. The term " lemma " can be translated as " word" or " main idea ". This indicates that it is a key idea, which is useful in many situations.

Examples

Famous lemmas

Lemmas often bear the name of its discoverer. Examples are:

• Lemma of Zorn
• Sperner 's Lemma
• Lemma of Euclid
• Lemma of Gauss
• Lemma Itō
• Lemma Zassenhaus

More examples can be found in the list of mathematical sets.

Example of the use of a lemma

Example, one can show that is irrational ( as a set ), if one can assume that the squares of even numbers are straight again, squares of odd numbers but always give odd numbers ( this statement correspond to the lemma). To proceed structured, we prove the two facts alone, although the fact of the lemma ( the lemma ) can later be applied to other cases or evidence, whereas the "sentence" a special statement provides.

To implement the previous example, you would ( for example in a lecture) as follows.

Lemma Squares of even and odd integers is always even or odd.

Proof: Let prescribed. We show that the corresponding statement is sufficient, ie, if (even) or (odd ) for a, then is even or odd.

Both cases are treated separately. In the first case () has (according to the rules of calculation power ), ie an even number. In the other case () results (after binomial formula ), ie an odd number.

Sentence is irrational, that is true.

Proof: The claimed statement is proved by the assumption is the opposite is true leads to a contradiction ( proof by contradiction ).

It is assumed, it is true. Then there are mutually prime with. Squaring this equation and multiplying both sides by, we get. Because the left side is straight, and the right straight. According to the previous lemma is then just ( because if odd, would be odd) and there is a with. From the equation, from which we see that and hence are ( again because of the lemma ) straight. This contradicts the assumption that and are elected prime. Thus, the assumption is rational, false, and the theorem is proved.

In the proof twice the previous lemma was used.