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''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.