# Proposition

A proposition or lemma ( ancient Greek λῆμμα lēmma , 'taking', 'assumption'; plural : “lemmas”) is a mathematical or logical statement that is used in the proof of a proposition , but which is not itself given the rank of a proposition. The distinction between sentences and lemmas is fluid and not objective. The term “lemma” can also be translated as “keyword” or “main idea”. This signals that it is a key thought that is useful in many situations.

## Examples

### Famous lemmas

Lemmas often bear the name of their discoverer. Examples for this are:

Further examples can be found in the list of mathematical theorems .

### Example of the use of a lemma

For example, one can show that is irrational (as a proposition) if one can assume that squares of even numbers are even again, but squares of odd numbers always result in odd numbers (this statement would correspond to the lemma). In order to proceed in a more structured manner, one proves the two facts individually, whereby the fact of the auxiliary proposition (the lemma) can later be applied to further cases or proofs, whereas the “proposition” provides a special statement. ${\ displaystyle {\ sqrt {2}}}$

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

Lemma: Squares of even and odd whole numbers are always even and odd, respectively.

Proof: be given. It has to be shown that the corresponding assertion is sufficient, ie if (even) or (odd) is for a , then is even or odd. ${\ displaystyle x \ in \ mathbb {Z}}$${\ displaystyle x ^ {2}}$${\ displaystyle x = 2y}$${\ displaystyle x = 2y + 1}$${\ displaystyle y \ in \ mathbb {Z}}$${\ displaystyle x ^ {2}}$

Both cases are dealt with separately. In the first case ( ) one has (according to the power calculation rules ) , i.e. an even number. In the other case ( ) the result is (according to the binomial formula ) , i.e. an odd number. ${\ displaystyle x = 2y}$${\ displaystyle x ^ {2} = (2y) ^ {2} = 2 ^ {2} \ cdot y ^ {2}}$${\ displaystyle = 2 \ cdot 2y ^ {2}}$${\ displaystyle x = 2y + 1}$${\ displaystyle x ^ {2} = (2y + 1) ^ {2} = (2y) ^ {2} +2 \ cdot 2y \ cdot 1 + 1 ^ {2}}$${\ displaystyle = 2 \ cdot (2y ^ {2} + 2y) +1}$

Theorem: is irrational, so it holds . ${\ displaystyle {\ sqrt {2}}}$${\ displaystyle {\ sqrt {2}} \ in \ mathbb {R} \ setminus \ mathbb {Q}}$

Proof: The asserted statement is proven by leading the assumption that the opposite is correct to a contradiction (contradiction proof).

It is believed to hold true . Then there are coprime to each other and with . If you square this equation and multiply both sides by , you get . Because the left side is straight, the right side is straight too. According to the previous lemma, then there is also even (because if it were odd, it would be odd) and there is a with . From the equation it follows from which one recognizes that and thus also (again because of the lemma) are straight. This contradicts the assumption that and have been chosen relatively prime. With that the assumption that is rational is wrong and the proposition is proven. ${\ displaystyle {\ sqrt {2}} \ in \ mathbb {Q}}$${\ displaystyle a \ in \ mathbb {Z}}$${\ displaystyle b \ in \ mathbb {N}}$${\ displaystyle {\ sqrt {2}} = {\ frac {a} {b}}}$${\ displaystyle b ^ {2}}$${\ displaystyle 2 \ cdot b ^ {2} = a ^ {2}}$${\ displaystyle a}$${\ displaystyle a}$${\ displaystyle a ^ {2}}$${\ displaystyle c \ in \ mathbb {Z}}$${\ displaystyle a = 2c}$${\ displaystyle b ^ {2} = {\ frac {a ^ {2}} {2}} = {\ frac {(2c) ^ {2}} {2}} = 2c ^ {2}}$${\ displaystyle b ^ {2}}$${\ displaystyle b}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle {\ sqrt {2}}}$

The previous lemma was used twice in the proof.

## Individual evidence

1. ^ Wilhelm Pape: Concise dictionary of the Greek language . Braunschweig, 1914, Volume 2, p. 39, keyword λῆμμα. ( at zeno.org )
2. Albrecht Beutelspacher : That is trivial oBdA! 2nd Edition. Vieweg Verlag, Wiesbaden 1992, ISBN 3-528-16442-5 , pp. 13 f .