What is "without loss of generality"

adv. (context mathematics English) With a constraining assumption that, however, makes it clear how to apply the proof performed under this assumption to the general case unconstrained by the assumption.

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. 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 ⇒ P(y,x) ⇒ P(x,y); hence, P(x,y) holds in all cases.