|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Correction to 'WLOG'
|
Misleading example of wolog by stevecheng
Correction id: 6909
Filed on: 2005-07-22 16:14:44
Status: Accepted on 2005-08-14 12:53:37
Type: Erratum
Correction text:
The example given:
``For example, we might be discussing properties of a segment (open or closed) of the real number line. Due to the nature of the reals, we can select endpoints $ a$ and $ b$ without loss of generality.''
is misleading, because that is not the situation where the wolog phrase is used ---
a segment of the real, by defintion, is a set of the form $(a,b)$ (or $[a,b]$ for closed sets etc. )
for some $a, b$ real (possibly also $-\infty, +\infty$).
I suggest replacing it with something like this:
``For example, we might want to prove something about open intervals $(a, b)$ of the real
number line. But the proof might become too tedious if $a$ and $b$ were arbitrary real numbers, so in the proof we simply assume that $a = 0$ and $b = 1$,
and \emph{without loss of generality}, the same arguments apply to general intervals $(a, b)$
Depending on the proof, the loss of generality might be accomplished by
translating and scaling the interval to $(0,1)$
\emph{before} carrying out the argument, and then translating and rescaling back to $(a, b)$
afterwards.''
Another possible example, which I think is probably easier to understand
for those who don't understand what wolog is in the first place:
``For example, we might want to prove something about open subsets of the real number line. But every open subset is a disjoint union of open intervals $(a, b)$,
and it may suffice to prove the proposition only for intervals $(a, b)$, and piece the results back together afterwards. So the proof may assume the open set in question is of the form $(a, b)$, \emph{without loss of generality}.
|
Comment from object owner akrowne:
I actually think the first one is easier to understand, since the selection 0 and 1 is very clear.
Also provided reference to proof of fundamental theorem theorem of arithmetic as an example. |
|
|
|
|
|
|
|
|
|
|
