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 before an assumption in a proof which narrows the premise to some special case; it implies that the proof for that case can be easily applied to all others, or that all other cases are equivalent or similar.[1] 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 is often enabled by the presence of symmetry. For example, if some property P(x,y) of real numbers is known to be symmetrical 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, we may assume "without loss of generality" that x ≤ y. There is then no loss of generality in that assumption: once the case x ≤ y ⇒ P(x,y) has been proved, the other case follows by y ≤ x ⇒[2] P(y,x) ⇒[3] P(x,y); hence, P(x,y) holds in all cases.

Example

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

If three objects are each painted either red or blue, then there must be two objects of the same color.

A proof:

Assume without loss of generality that the first object is red. If either of the other two objects is red, we are finished; if not, the other two objects must both be blue and we are still finished.

This works because exactly the same reasoning (with "red" and "blue" interchanged) could be applied if the alternative assumption were made, namely that the first object is blue.

See also

References

  1. ↑ Chartrand, Gary; Polimeni, Albert D.; Zhang, Ping (2008), Mathematical Proofs / A Transition to Advanced Mathematics (2nd ed.), Pearson/Addison Wesley, pp. 80–81, ISBN 0-321-39053-9
  2. ↑ from the just proved implication by interchanging x and y
  3. ↑ by symmetry of P
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