''Without loss of generality'' (often
abbreviated
An abbreviation (from Latin ''brevis'', meaning ''short'') is a shortened form of a word or phrase, by any method. It may consist of a group of letters or words taken from the full version of the word or phrase; for example, the word ''abbrevi ...
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
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. The term is used to indicate the assumption that follows is chosen arbitrarily, narrowing the premise to a particular case, but does not affect the validity of the
proof in general. The other cases are sufficiently similar to the one presented that proving them follows by essentially the same logic. As a result, once a proof is given for the particular case, it is
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. For example, if some property ''P''(''x'',''y'') of
real numbers 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''
⇒ ''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
In logic and mathematics, proof by example (sometimes known as inappropriate generalization) is a logical fallacy whereby the validity of a statement is illustrated through one or more examples or cases—rather than a full-fledged proof.
The ...
– a
logical fallacy of proving a claim by proving a non-representative example.
Example
Consider the following
theorem (which is a case of the
pigeonhole principle
In mathematics, the pigeonhole principle states that if items are put into containers, with , then at least one container must contain more than one item. For example, if one has three gloves (and none is ambidextrous/reversible), then there mu ...
):
A proof:
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
*
Up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
*
Mathematical jargon
The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in ...
References
External links
*{{PlanetMath , urlname=WLOG, title=WLOG
"Without Loss of Generality" by John Harrison - discussion of formalizing "WLOG" arguments in an automated theorem prover.
Mathematical terminology