Without any loss of generality

  • #1
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. :)
 

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  • #2
phinds
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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?
 
  • #3
jbriggs444
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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.
 
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  • #4
i understand thanks!
 
  • #5
HallsofIvy
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Sometimes abbreviated "wlog".
 
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