Difference between revisions of "Without loss of generality"

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== Definition (Inspired from Wikipedia) ==
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==Definition==
Without loss of generality, often abbreviated to WLOG, is a frequently used expression in maths. The term is used to indicate that the following proof emphasizes on a particular case, but doesn’t affect the validity of the proof in general.
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Without loss of generality, often abbreviated to WLOG, is a frequently used expression in math. The term is used to indicate that the following proof emphasizes on a particular case, but doesn’t affect the validity of the proof in general.
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==Example==
  
== Example (from Wikipedia) ==
 
 
* If three objects are each painted either red or blue, then there must be at least two objects of the same color.
 
* If three objects are each painted either red or blue, then there must be at least two objects of the same color.
 
<math>\textbf{Proof}</math>:
 
<math>\textbf{Proof}</math>:
  
Assume, <math>\textbf{without loss of generality}</math>, that the first object is red. If either of the other two objects is red, then we are finished; if not, then the other two objects must both be blue and we are still finished.
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Assume, '''without loss of generality''', that the first object is red. If either of the other two objects is red, then we are finished; if not, then the other two objects must both be blue and we are still finished.
  
 
The above argument works because the exact same reasoning could be applied if the first object is blue. As a result, the use of "without loss of generality" is valid in this case.
 
The above argument works because the exact same reasoning could be applied if the first object is blue. As a result, the use of "without loss of generality" is valid in this case.
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(Note that this can also be proved by the [[Pigeonhole Principle]])
  
 
== Problems using WLOG ==
 
== Problems using WLOG ==
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Revision as of 19:33, 30 January 2023


Definition

Without loss of generality, often abbreviated to WLOG, is a frequently used expression in math. The term is used to indicate that the following proof emphasizes on a particular case, but doesn’t affect the validity of the proof in general.

Example

  • If three objects are each painted either red or blue, then there must be at least two objects of the same color.

$\textbf{Proof}$:

Assume, without loss of generality, that the first object is red. If either of the other two objects is red, then we are finished; if not, then the other two objects must both be blue and we are still finished.

The above argument works because the exact same reasoning could be applied if the first object is blue. As a result, the use of "without loss of generality" is valid in this case. (Note that this can also be proved by the Pigeonhole Principle)

Problems using WLOG

Read more

https://en.wikipedia.org/wiki/Without_loss_of_generality

https://www.cl.cam.ac.uk/~jrh13/papers/wlog.pdf


This article is a stub. Help us out by expanding it.