Without loss of generality

Without loss of generality , abbreviated o. B. d. A. , is a phrase found in mathematical proofs . In addition, the formulation is also used without restriction of generality ( o. E. d. A. ) or briefly without limitation ( oE or as a ligature Π).

These formulations express that a restriction (e.g. the range of values ​​of a variable ) is only assumed to simplify the argumentation (in particular to reduce the amount of paperwork) without affecting the validity of the statements made below with regard to the general public suffer from. The proof is only given for one of several possible cases. This happens under the condition that the other cases can be proven in an analogous way (e.g. with symmetry).

By o. B. d. A. trivial special cases can also be ignored.

example

Bolzano's interim value theorem

Theorem: A function continuous in the interval with the property · has in at least one zero. ${\displaystyle [a,b]}$ ${\displaystyle f}$ ${\displaystyle f(a)}$ ${\displaystyle f(b)<0}$ ${\displaystyle [a,b]}$

Proof: From · it follows that and are not zero and have different signs. O. B. d. A. let's consider the case and . ${\displaystyle f(a)}$ ${\displaystyle f(b)<0}$ ${\displaystyle f(a)}$ ${\displaystyle f(b)}$ ${\displaystyle f(a)<0}$ ${\displaystyle f(b)>0}$

... (for this case the proof follows)

It can be seen that in this argument the other case and is also covered by simply by replacing. That the generality is not restricted by this follows from three properties: ${\displaystyle f(a)>0}$ ${\displaystyle f(b)<0}$ ${\displaystyle f}$ ${\displaystyle -f}$

1. Is steady, then too . ${\displaystyle f}$ ${\displaystyle -f}$
2. If the function values ​​of at the interval limits are not zero and have different signs, then this also applies to . ${\displaystyle f}$ ${\displaystyle -f}$
3. The zeros of and match. ${\displaystyle f}$ ${\displaystyle -f}$

literature

• Albrecht Beutelspacher : That is o. B. d. A. trivial! . Vieweg + Teubner Verlag, 9th edition (2009), ISBN 3-834-80771-0

Web links

Wiktionary: o. B. d. A.  - Explanations of meanings, word origins, synonyms, translations