Without loss of generality
Without loss of generality(oftenabbreviatedtoWOLOG,WLOGorw.l.o.g.;less commonly stated aswithout any loss of generalityorwith no loss of generality) is a frequently used expression inmathematics.The term is used to indicate the assumption that what follows is chosen arbitrarily, narrowing the premise to a particular case, but does not affect the validity of theproofin general. The other cases are sufficiently similar to the one presented that proving them follows by essentially the same logic.[1]As a result, once a proof is given for the particular case, it istrivialto adapt it to prove the conclusion in all other cases.
In many scenarios, the use of "without loss of generality" is made possible by the presence ofsymmetry.[2]For example, if some propertyP(x,y) ofreal numbersis known to be symmetric inxandy,namely thatP(x,y) is equivalent toP(y,x), then in proving thatP(x,y) holds for everyxandy,one may assume "without loss of generality" thatx≤y.There is no loss of generality in this assumption, since once the casex≤y⇒P(x,y) has been proved, the other case follows by interchangingxandy :y≤x⇒P(y,x), and by symmetry ofP,this impliesP(x,y), thereby showing thatP(x,y) holds for all cases.
On the other hand, if neither such a symmetry nor another form of equivalence can be established, then the use of "without loss of generality" is incorrect and can amount to an instance ofproof by example– alogical fallacyof proving a claim by proving a non-representative example.[3]
Example
[edit]Consider the followingtheorem(which is a case of thepigeonhole principle):
If three objects are each painted either red or blue, then there must be at least two objects of the same color.
A 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 alternative assumption, namely, that the first object is blue, were made, or, similarly, that the words 'red' and 'blue' can be freely exchanged in the wording of the proof. As a result, the use of "without loss of generality" is valid in this case.
See also
[edit]References
[edit]- ^Chartrand, Gary;Polimeni, Albert D.;Zhang, Ping(2008).Mathematical Proofs / A Transition to Advanced Mathematics(2nd ed.). Pearson/Addison Wesley. pp. 80–81.ISBN978-0-321-39053-0.
- ^Dijkstra, Edsger W.(1997). "WLOG, or the misery of the unordered pair (EWD1223)". In Broy, Manfred; Schieder, Birgit (eds.).Mathematical Methods in Program Development(PDF).NATO ASI Series F: Computer and Systems Sciences. Vol. 158. Springer. pp. 33–34.doi:10.1007/978-3-642-60858-2_9.ISBN978-3-642-64588-4.
- ^"An Acyclic Inequality in Three Variables".www.cut-the-knot.org.Retrieved2019-10-21.