Definitions for WithOut Loss of Generality
with·out loss of gen·er·al·i·ty

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1. without loss of generalityadverb

   Making a constraining assumption that makes it clear how to apply the proof performed under this assumption to the general case unconstrained by the assumption.

WikipediaRate this definition:0.0 / 0 votes

1. Without loss of generality

   Without loss of generality (often 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 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 – a logical fallacy of proving a claim by proving a non-representative example.

WikidataRate this definition:3.3 / 3 votes

1. Without loss of generality

   Without loss of generality is a frequently used expression in mathematics. The term is used before an assumption in a proof which narrows the premise to some special case; it is implied that the proof for that case can be easily applied to all others, or that all other cases are equivalent. Thus, given a proof of the conclusion in the special case, it is trivial to adapt it to prove the conclusion in all other cases. This often requires the presence of symmetry. For example, in proving, if we wish to assume "without loss of generality" that, then it is required that be symmetrical in and, namely that is equivalent to . There is then no loss of generality in assuming, since a proof for that case can trivially be adapted for the other case by interchanging and

How to pronounce WithOut Loss of Generality?

1. Alex

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How to say WithOut Loss of Generality in sign language?

Numerology

1. Chaldean Numerology

   The numerical value of WithOut Loss of Generality in Chaldean Numerology is: 4

2. Pythagorean Numerology

   The numerical value of WithOut Loss of Generality in Pythagorean Numerology is: 3

References

1. ^ Wiktionary
   https://en.wiktionary.org/wiki/Without_Loss_of_Generality
2. ^ Wikipedia
   https://en.wikipedia.org/wiki/Without_Loss_of_Generality
3. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=WithOut Loss of Generality