Difference between revisions of "Without loss of generality"
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| − | == Definition | + | ==Definition== |
| − | Without loss of generality, often abbreviated to WLOG, is a frequently used expression in | + | |
| + | 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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| + | Be careful when using WLOG in a proof. By using it, you must be certain that your statement actually DOES work for all cases! | ||
| + | If you use WLOG in a proof and the statement is not necessarily true, points will get marked off. For example, you can't say "WLOG, let <math>a > b > c</math>." if <math>a</math> could equal <math>b</math> or <math>c</math>. | ||
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| + | For introductory and intermediate competitions such as the AMC 10, AMC 12, and AIME competitions it is very common to use WLOG in geometry problems by assuming one undefined side length to be equal to 1. | ||
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| + | ==Example== | ||
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* 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, | + | 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. | ||
| + | (Note that this can also be proved by the [[Pigeonhole Principle]]) | ||
== Problems using WLOG == | == Problems using WLOG == | ||
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* [[2016_AMC_12A_Problems/Problem_17 | 2016 AMC 12A Problem 17]] (See Solution 2) | * [[2016_AMC_12A_Problems/Problem_17 | 2016 AMC 12A Problem 17]] (See Solution 2) | ||
* [[2012_AMC_10A_Problems/Problem 23 | 2012 AMC 10A Problem 23]] | * [[2012_AMC_10A_Problems/Problem 23 | 2012 AMC 10A Problem 23]] | ||
| + | * [[2018_AMC_12B_Problems/Problem 18 | 2018 AMC 10B Problem 18]] | ||
| + | * [[2005_Austrian_Mathematical_Olympiad_Final_Round-Part 1/Problem 5 | 2005 Austrian MO Final Round-Part 1 Problem 5]] | ||
== Read more == | == Read more == | ||
Latest revision as of 22:35, 13 September 2024
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.
Be careful when using WLOG in a proof. By using it, you must be certain that your statement actually DOES work for all cases!
If you use WLOG in a proof and the statement is not necessarily true, points will get marked off. For example, you can't say "WLOG, let 
." if 
could equal 
or 
.
For introductory and intermediate competitions such as the AMC 10, AMC 12, and AIME competitions it is very common to use WLOG in geometry problems by assuming one undefined side length to be equal to 1.
Example
- If three objects are each painted either red or blue, then there must be at least two objects of the same color.
:
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
- 2017 USAJMO Problem 3
- 2016 AMC 12A Problem 17 (See Solution 2)
- 2012 AMC 10A Problem 23
- 2018 AMC 10B Problem 18
- 2005 Austrian MO Final Round-Part 1 Problem 5
Read more
https://en.wikipedia.org/wiki/Without_loss_of_generality
https://www.cl.cam.ac.uk/~jrh13/papers/wlog.pdf
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