Without any loss of generality

[Kostas Tzim]

Hello guys can someone please explain to me how to use the assertion, "without any loss of generality assume...", i find it kind of tricky to use...a simple example would be useful too. :)

 

[phinds]

Hello guys can someone please explain to me how to use the assertion, "without any loss of generality assume...", i find it kind of tricky to use...a simple example would be useful too. :)

Context ? Assume WHAT?

 

[jbriggs444]

Hello guys can someone please explain to me how to use the assertion, "without any loss of generality assume...", i find it kind of tricky to use...a simple example would be useful too. :)

Tossing out an example: If a and b are distinct real numbers prove that a^2 + b^2 > 2ab.

Proof:

Without loss of generality, assume a > b [... rest of proof ensues ...]

We can make this assumption because a and b are distinct. Either a > b or b > a. If a > b then our assumption is true. If b > a then we could reverse the labels and the re-labelled assumption is true. The rest of the proof would go through either way. There is no point in writing essentially the same proof twice, once with labels a and b and then again with labels b and a.

The "without loss of generality" phrasing is used to indicate that this sort of situation exists -- that all of the cases under consideration are really just re-labellings of a single base case and that no other possibilities exist.

 

[Kostas Tzim]

i understand thanks!

 

[HallsofIvy]

Sometimes abbreviated "wlog".