Editing Without loss of generality

{{Short description|Expression in mathematics}}
{{redirect|WLOG|the radio station|WLOG (FM)}}
'''''Without loss of generality''''' (often [[Acronym|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 to indicate the assumption that what follows is chosen arbitrarily, narrowing the premise to a particular case, but does not affect the validity of the [[mathematical proof|proof]] in general. The other cases are sufficiently similar to the one presented that proving them follows by essentially the same logic.<ref>{{cite book|first1=Gary|last1=Chartrand|author1-link=Gary Chartrand|first2=Albert D.|last2=Polimeni|first3=Ping|last3=Zhang|author3-link=Ping Zhang (graph theorist)|title=Mathematical Proofs / A Transition to Advanced Mathematics|edition=2nd|publisher=Pearson/Addison Wesley|year=2008|isbn=978-0-321-39053-0|pages=80–81}}</ref> As a result, once a proof is given for the particular case, it is [[Trivial (mathematics)|trivial]] to 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 of [[symmetry]].<ref>{{cite book | last = Dijkstra | first = Edsger W. | author-link = Edsger W. Dijkstra | editor1-last = Broy | editor1-first = Manfred | editor2-last = Schieder | editor2-first = Birgit | contribution = WLOG, or the misery of the unordered pair (EWD1223) | doi = 10.1007/978-3-642-60858-2_9 | pages = 33–34 | publisher = Springer | series = NATO ASI Series F: Computer and Systems Sciences | title = Mathematical Methods in Program Development | url = https://www.cs.utexas.edu/~EWD/ewd12xx/EWD1223.PDF | volume = 158 | year = 1997| isbn = 978-3-642-64588-4 }}</ref> For example, if some property ''P''(''x'',''y'') of [[real number]]s is known to be symmetric in ''x'' and ''y'', namely that ''P''(''x'',''y'') is equivalent to ''P''(''y'',''x''), then in proving that ''P''(''x'',''y'') holds for every ''x'' and ''y'', one may assume "without loss of generality" that ''x'' ≤ ''y''. There is no loss of generality in this assumption, since once the case ''x'' ≤ ''y'' [[Material conditional|⇒]] ''P''(''x'',''y'') has been proved, the other case follows by interchanging ''x'' and ''y'': ''y'' ≤ ''x'' ⇒ ''P''(''y'',''x''), and by symmetry of ''P'', this implies ''P''(''x'',''y''), thereby showing that ''P''(''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 of [[proof by example]] – a [[logical fallacy]] of proving a claim by proving a non-representative example.<ref>{{Cite web|url=https://www.cut-the-knot.org/m/Algebra/AcyclicInequalityInThreeVariables.shtml|title=An Acyclic Inequality in Three Variables|website=www.cut-the-knot.org|access-date=2019-10-21}}</ref>

== Example ==

Consider the following [[theorem]] (which is a case of the [[pigeonhole principle]]):

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

A proof: 
{{quote|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.|sign=|source=}}

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==
* [[Mathematical jargon]]
* [[Up to]]

==References==
{{reflist}}

==External links==
*{{PlanetMath |urlname=WLOG|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]]