Without loss of generality
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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 before an assumption in a proof which does not narrow the premise of the proof.
WLOG means that the conclusions of the proof for the premise follow from a subproof on part of the 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, so if P(x, y) is true or false, and P(x*, y*) is true for some x*, y*, then P(y*, x*) is also true. (We then say that P is symmetric or commutative).
Next to WLOG there must be present an assumption. To check that there is no loss of generality, write out the entire proof (without making the simplifying assumption) and then see if the proof which you wrote out follows out of a proof of just a part of the premise.
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 begin the full proof by listing all the permutations:
- RRR
- RRB
- RBR
- BRR
- RBB
- BRB
- BBR
- BBB
of which there are eight, as we expect (2 × 2 × 2). We now see that the separated lists are equivalent under our assumptions, so we reduce the list:
- {R, R, R}
- {R, B, R}
- {R, B, B}
- {B, B, B}
Now we scan the shorter list with our eyes and see that there are two objects of the same color in every case.
We reduced the premise as we did by noticing, first, that the order of tje objects doesn't matter; second, that we are interested not in the kind of color but in its count. Both assumptions are consistent with our conclusion, which required that two objects be of the same color -- a loose requirement.