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'''Without loss of generality''' (often abbreviated to '''WOLOG''', '''WLOG''' or '''w.l.o.g.'''; less commonly stated as '''without any loss of generality''' or '''with no loss of generality''') is a frequently used expression in [[mathematics]]. The term is used before an assumption in a proof which narrows the premise to some special case; it is implied that the proof for that case can be easily applied to all others, or that all other cases are equivalent or similar.<ref>{{citation|first1=Gary|last1=Chartrand|first2=Albert D.|last2=Polimeni|first3=Ping|last3=Zhang|title=Mathematical Proofs / A Transition to Advanced Mathematics|edition=2nd|publisher=Pearson/Addison Wesley|year=2008|isbn=0-321-39053-9|pages=80-81}}</ref> Thus, given a proof of the conclusion in the special case, it is [[Trivial (mathematics)|trivial]] to adapt it to prove the conclusion in all other cases. | |||
This often requires the presence of [[symmetry]]. For example, in proving <math>P(x,y)</math> (''[[i.e.]]'', that some property <math>P</math> holds for any two [[real number]]s <math>x</math> and <math>y</math>), if we wish to assume "without loss of generality" that <math>x\leq y</math>, then it is required that <math>P</math> be symmetrical in <math>x</math> and <math>y</math>, namely that <math>P(x,y)</math> is equivalent to <math>P(y,x)</math>. There is then no loss of generality in assuming <math>x\leq y</math>, since a proof for that case can trivially be adapted for the other case <math>(y\leq x)</math> by interchanging <math>x</math> and <math>y</math> (leading to the conclusion <math>P(y,x)</math>, which is known to be equivalent to <math>P(x,y)</math>, the desired conclusion.) | |||
== Example == | |||
Consider the following [[theorem]] (which is a case of the [[pigeonhole principle]]): | |||
{{bquote|If three objects are each painted either red or blue, then there must be two objects of the same color.}} | |||
A proof: | |||
{{bquote|Assume without loss of generality that the first object is red. If either of the other two objects is red, we are finished; if not, the other two objects must both be blue and we are still finished.}} | |||
This works because exactly the same reasoning (with "red" and "blue" interchanged) could be applied if the alternative assumption were made, namely that the first object is blue. | |||
==See also== | |||
* [[up to]] | |||
* [[mathematical jargon]] | |||
==References== | |||
{{reflist}} | |||
==External links== | |||
*{{planetmath reference|id=2946|title=WLOG}} | |||
*[http://www.cl.cam.ac.uk/~jrh13/papers/wlog.pdf "Without Loss of Generality" by John Harrison - discussion of formalizing "WLOG" arguments in an automated theorem prover.] | |||
[[Category:Mathematical terminology]] | |||
Revision as of 22:42, 15 January 2014
Without loss of generality (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as without any loss of generality or with no loss of generality) is a frequently used expression in mathematics. The term is used before an assumption in a proof which narrows the premise to some special case; it is implied that the proof for that case can be easily applied to all others, or that all other cases are equivalent or similar.[1] Thus, given a proof of the conclusion in the special case, it is trivial to adapt it to prove the conclusion in all other cases.
This often requires the presence of symmetry. For example, in proving (i.e., that some property holds for any two real numbers and ), if we wish to assume "without loss of generality" that , then it is required that be symmetrical in and , namely that is equivalent to . There is then no loss of generality in assuming , since a proof for that case can trivially be adapted for the other case by interchanging and (leading to the conclusion , which is known to be equivalent to , the desired conclusion.)
Example
Consider the following theorem (which is a case of the pigeonhole principle):
A proof: Template:Bquote
This works because exactly the same reasoning (with "red" and "blue" interchanged) could be applied if the alternative assumption were made, namely that the first object is blue.
See also
References
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External links
- Template:Planetmath reference
- "Without Loss of Generality" by John Harrison - discussion of formalizing "WLOG" arguments in an automated theorem prover.
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