''Without loss of generality'' (often
abbreviated
An abbreviation () is a shortened form of a word or phrase, by any method including shortening, contraction, initialism (which includes acronym), or crasis. An abbreviation may be a shortened form of a word, usually ended with a trailing per ...
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 a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
. 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
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a co ...
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
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
. For example, if some property ''P''(''x'',''y'') of
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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''
⇒ ''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
In logic and philosophy, a formal fallacy is a pattern of reasoning rendered invalid by a flaw in its logical structure. Propositional logic, for example, is concerned with the meanings of sentences and the relationships between them. It focuses ...
of proving a claim by proving a non-representative example.
Example
Consider the following
theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
(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, of three gloves, at least two must be right-handed or at least two must be l ...
):
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
*
Mathematical jargon
The language of mathematics has a wide 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 ...
*
Up to Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation "
* if and are related by , that is,
* if holds, that is,
* if the equivalence classes of and with respect to are equal.
This figure of speech ...
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