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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.
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| 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.)
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| == Example ==
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| Consider the following [[theorem]] (which is a case of the [[pigeonhole principle]]):
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| {{bquote|If three objects are each painted either red or blue, then there must be two objects of the same color.}}
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| A proof:
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| {{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.}}
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| 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.
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| ==See also==
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| * [[up to]]
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| * [[mathematical jargon]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *{{planetmath reference|id=2946|title=WLOG}}
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| *[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.]
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| [[Category:Mathematical terminology]]
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Here is my web-site: clash of clans hacker.exe