Without loss of generality
From Exampleproblems
Without loss of generality (abbreviated to WLOG or WOLOG and less commonly stated as without any loss of generality) is a frequently used expression in mathematics. The term is used where the situation or situations described can be trivially generalized to all needed situations.
WLOG is not a mathematical axiom of any kind; it is an admission to the reader that the proof is not complete, but trivially completable by enumerating missing cases or applying an unstated theorem that will allow us to generalize a proof on some subset of objects to the set of all such objects. As with all such shorthands, there is always a danger of introducing a false implicit assumption (in other words, that there is in fact loss of generality we have missed by not writing out the proof in full).
Example
Consider the following theorem (the simplest case of Ramsey's theorem and also an example of Dirichlet's pigeonhole principle):
Three objects are each painted either red or blue; there must be two objects of the same color.
The 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.
We can assume WLOG that the first object is red, because there is no difference between red and blue for the purposes of the proof. If the first object is blue instead of red, that is equivalent to a mere change of the names of the two colors, and the names of the colors don't matter; the proof goes through just fine if you switch 'red' to 'blue' and vice versa.
