Without loss of generality (abbreviated to WLOG; 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 is implied that the proof on this subset can be easily applied to all others (or that all other cases are trivial). Thus, given a proof of the special case, it is trivial to show that the conclusions follow from the full premise.
This often requires the presence of symmetry. For example, if two numbers are called x, y, and it is known that x < y, then any relationship proved based on this assumption will hold for the complementary relation, y < x, because the roles of x and y are interchanged, but the proof is symmetric in the two variables. In other words, if we know that P(x, y) is true if and only if P(y, x) is true, then without loss of generality it is enough to show P(x, y) is true (since P(y, x) then immediately follows, by symmetry). (In this context, we call P symmetric.)
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.